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This article is cited in 5 scientific papers (total in 5 papers) On stabilization of solutions of second-order semilinear parabolic equations on closed manifolds D. V. Tunitsky V. A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, Moscow
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     IntroductionSemilinear second-order parabolic equations of the form where is a linear elliptic differential operator, proved widely useful in mathematical modeling of various reaction–diffusion processes. Among numerous studies on these equations, we mention, first of all, the famous papers by Kolmogorov, Petrovskii, Piskunov [1] and Fisher [2], which dealt with a homogeneous operator L and a homogeneous right-hand side f, We also mention the paper [3], which provides a historical account and a bibliographical survey on the studies of equations (0.1), (0.2). Some applications of these equations can be found in the book [4].Equations (0.1), (0.2) in which the operator L has periodic coefficients were studied in [3]. This case, which is reduced to an equation on a manifold diffeomorphic to the n-dimensional torus Tn, has great applied value. Of special importance here are analogues of equations (0.1), (0.2) on other closed manifolds, and, in particular, on those diffeomorphic the n-dimensional sphere Sn. In this regard, it is worth pointing out that, according to the generalized Poincaré conjecture, the sphere Sn is homeomorphic to each closed manifold homotopically equivalent to Sn. The greatest challenge in this conjecture is in the dimension n=3. In this case, the proof proposed by G. Ya. Perelman is based on the study of Ricci flows on closed three-dimensional manifolds (see [5]–[7]). This very approach proved capable of justifying the Poincaré conjecture, thereby solving one of the “millennium problems”. With the help of the DeTurck trick (see [8]), the problem of finding Ricci flows can be essentially reduced to solution of the corresponding parabolic equations. This clearly demonstrates that the study of solutions of nonlinear parabolic equations on closed manifolds has great applied and mathematical value. In the present paper, we consider existence, uniqueness, and stabilization of solutions of analogues of equation (0.1) on arbitrary closed finite-dimensional manifolds. It is worth pointing out that, in many applied problems (for example, in many control problems), the right-hand sides of equations (0.1), (0.2) may involve terms which are not smooth or even not continuous. So, it is desirable to be able to select a class of admissible solutions based on which a satisfactory theory of solvability of such equations can be constructed under minimal requirements on the regularity of their coefficients. In the present paper, the weak solutions are considered as such a class. In this class, it proves possible to investigate solvability of analogues of the inhomogeneous equation (0.1) on closed manifolds under fairly light requirements on the regularity of their coefficients. In particular, some coefficients of these equations can be generalized function. § 1. Statement of the problem1.1. Function spaces of tensor fieldsBy X we denote an n-dimensional smooth closed Riemannian manifold, that is, a connected Hausdorff compact manifold without boundary equipped with metric g:TX×TX→R. The metrics induced by g on tensor bundles (TX)⊗m⊗(T∗X)⊗l, m,l=0,1,2,…, of this manifold will be denoted by the same letter g; further, by (TX)⊗0⊗(T∗X)⊗0 we will denote the trivial bundle X×R with g(r,t)=rt for r,t∈R. The metric g induces on the manifold X the measure V=Vg in the local coordinates by x1,…,xn where g=det(g(∂/∂xm,∂/∂xl)), and the Levi-Civita connection with the covariant differentiation operator ∇=∇g uniquely defined by this connection.Given real-value functions u and v on X, we set From the metric g and the measure V, one defines, in the usual way, the function space Lp(X) and the space of tensor fields Lp((TX)⊗m⊗(T∗X)⊗l), where p⩾ and m,l=0,1,2,\dots . Similarly, with the help of the covariant differentiation operator \nabla, one defines the Sobolev spaces W^{k,p}(X) and W^{k,p}\bigl((TX)^{\otimes^m}\otimes (T^* X)^{\otimes^l}\bigr), k=0,1,2,\dots, and the Hölder spaces C^{k,\alpha}(X) and C^{k,\alpha}\bigl((TX)^{\otimes^m}\otimes(T^* X)^{\otimes^l}\bigr) for 0<\alpha \leqslant 1 (see [9], § 10.2.4, and [10], § 1).On the tangent bundle of the Cartesian product [0,T)\times X, where T \in (0,+\infty], consider the metric
\begin{equation*}
g_{[0,T)\times X}\colon T([0,T)\times X)\ni (\tau,\xi)\mapsto \tau^2+g(\xi,\xi)\in \mathbb{R}
\end{equation*}
\notag
and the corresponding measure V_{g_{[0,T)\times X}} and the covariant differentiation operator \nabla_{g_{[0,T)\times X}}, using which, we similarly construct the function spaces
\begin{equation*}
\begin{gathered} \, L^p([0,T)\times X),\qquad L^p\bigl(\bigl(T([0,T) \times X)\bigr)^{\otimes^m} \otimes \bigl(T^*([0,T)\times X)\bigr)^{\otimes^l}\bigr), \\ W^{k,p}([0,T)\times X),\qquad W^{k,p}\bigl(\bigl(T([0,T) \times X)\bigr)^{\otimes^m} \otimes \bigl(T^*([0,T)\times X)\bigr)^{\otimes^l}\bigr), \\ C^{k,\alpha}([0,T)\times X),\qquad C^{k,\alpha} \bigl(\bigl(T([0,T) \times X)\bigr)^{\otimes^m} \otimes \bigl(T^*([0,T)\times X)\bigr)^{\otimes^l}\bigr). \end{gathered}
\end{equation*}
\notag
If B is a Banach space with norm {\|\,{\cdot}\,\|_B}, then, for T \in (0,+\infty], one defines in the standard way, the Banach spaces L^p(T;B) with the norms
\begin{equation*}
\|q\|_{L^p(T;B)}=\biggl(\int_0^T\|q(t)\|_B^p\,dt\biggr)^{1/p}, \quad p\geqslant 1, \qquad \|q\|_{L^\infty(T;B)}= \operatorname*{ess\,sup}_{t\in [0,T)}\|q(t)\|_B,
\end{equation*}
\notag
see Chap. III, § 1 in [11] and Chap. II, § 2 in [12]. The spaces L^2(T;W^{1,2}(X)) and W^{1,2}(T;L^2(X)) are Hilbert spaces equipped with the inner products
\begin{equation*}
\begin{aligned} \, \langle q,p\rangle_{L^2(T;W^{1,2}(X))}&= \int_0^T\langle q(t),p(t)\rangle_{W^{1,2}(X)}\,dt, \\ \langle q,p\rangle_{W^{1,2}(T;L^2(X))}&= \int_0^T\bigl(\langle q(t),p(t)\rangle+ \langle q'(t),p'(t)\rangle\bigr)\,dt; \end{aligned}
\end{equation*}
\notag
their intersection L^2(T;W^{1,2}(X))\cap W^{1,2}(T;L^2(X)) is isomorphic to the Hilbert space W^{1,2}([0,T)\times X) with the inner product
\begin{equation*}
\langle q,p\rangle_{W^{1,2}([0,T)\times X)}= \int_0^T\bigl(\langle q(t),p(t)\rangle+\langle dq(t),dp(t)\rangle_{L^2(T^* X)} +\langle q'(t),p'(t)\rangle\bigr)\,dt.
\end{equation*}
\notag
Here and in what follows, q(t,\,\cdot\,) is identified with q(t). Setting we get a Banach space with the norm
\begin{equation*}
\|q\|_{W(T;X)}^2=\operatorname*{ess\,sup}_{t\in [0,T)} \langle q(t),q(t)\rangle+\int_0^T\langle dq(t),dq(t)\rangle_{L^2(T^* X)}\,dt.
\end{equation*}
\notag
As usual, for a linear semilocal subspace E \subseteq \mathcal{D}'([0,T)\times X), we let E_{\mathrm{loc}} denote the smallest local subspace \mathcal{D}'([0,T)\times X) containing E (see § 10.1 in [13]). In particular, for L^2([0,T)\times X) we have L_{\mathrm{loc}}^2([0,T)\times X), for L^\infty([0,T)\times X) we have L_{\mathrm{loc}}^\infty([0,T)\times X), and for W(T;X) we have W_{\mathrm{loc}}(T;X). 1.2. Elliptic equationsIn what follows, a particular property is said to hold almost everywhere if the set of points for which the property fails to hold is a set of measure zero with respect to the measure V (see (1.1)). Let the Riemannian manifold X be equipped, in addition to g, with another metric a. Assume that this metric is measurable, and there exist positive numbers a_0 and a_1 such that, almost everywhere,
\begin{equation}
a_0 g(\eta,\eta) \leqslant a(\eta,\eta) \leqslant a_1 g(\eta,\eta)
\end{equation}
\tag{1.2}
for all \eta \in T^* X. Consider the operators d_a^* and d_g^* formally adjoint with the exterior differentiation operator d with respect to the metrics a and g, respectively (see [14], Chap. VIII, § 1). In particular, \langle a(du,v),1\rangle=\langle a(u,d_a^* v),1\rangle for all differential k-forms u and (k+1)-forms v, k=0,1,\dots,n-1, and if the manifold X is orientable, then, d_a^*=(-1)^{n(k+1)+1}*d\,* on k-forms, where * = *_a is the Hodge operator induced by the metric a. Given a function u\in C^\infty(X), we define the linear second-order differential operator where \Delta=\Delta_a=d_a^*\circ d is the geometric Laplacian (the Laplace–de Rahm operator; see Chap. IV, § 5 in [15]), and b, c are measurable bounded (with respect to the metric g) vector field and linear differential form on X, respectively. In the local coordinates x^1,\dots,x^n,
\begin{equation*}
\begin{aligned} \, Lu&=-\frac{1}{\sqrt{a}} \sum_{l,m=1}^n \frac{\partial}{\partial x^l} \biggl(\sqrt{a}\,a(dx^l,dx^m)\, \frac{\partial u}{\partial x^m}\biggr)+ \sum_{l=1}^n b(dx^l)\,\frac{\partial u}{\partial x^l} \\ &\qquad-\frac{1}{\sqrt{g}} \sum_{l,m=1}^n \frac{\partial}{\partial x^l} \biggl(\sqrt{g}\,g(dx^l,dx^m) uc\biggl(\frac{\partial}{\partial x^m}\biggr)\biggr), \end{aligned}
\end{equation*}
\notag
where a=\det\bigl(a(\partial/\partial x^m,\partial/\partial x^l)\bigr) and g=\det\bigl(g(\partial/\partial x^m,\partial/\partial x^l)\bigr); cf. (0.2). In this context, condition (1.2) means that operator (1.3) is uniformly elliptic on the manifold X. Consider the function
\begin{equation}
f\colon X \times \mathbb{R} \ni (x,r)\mapsto f(x,r)\in \mathbb{R}
\end{equation}
\tag{1.4}
which is locally Lipschitz continuous almost everywhere in r, that is, for any r\in \mathbb{R} there exists a positive constant \mu_0=\mu_0(r) such that, for r_1,r_2 \in [-r,r],
\begin{equation}
|f(\,\cdot\, ,r_1)-f(\,\cdot\, ,r_2)| \leqslant \mu_0(r)|r_1-r_2|
\end{equation}
\tag{1.5}
almost everywhere. A weak (or generalized) solution of the equation where h is a measurable bounded linear differential form on the manifold X, is a function u \in W^{1,2}(X) such that f(\,\cdot\, ,u) \in L^2(X), and
\begin{equation}
\mathcal{L}(u,v)=\langle f(\,\cdot\, ,u),v\rangle+ \langle h,dv\rangle_{L^2(T^* X)}
\end{equation}
\tag{1.7}
for each function v \in C^\infty(X), where \mathcal{L} is a continuous bilinear form
\begin{equation}
\mathcal{L}\colon W^{1,2}(X) \times C^\infty(X) \ni (u,v)\mapsto \langle a(du,dv),1\rangle+\langle bu,v\rangle+ \langle uc,dv\rangle_{L^2(T^* X)} \in \mathbb{R}.
\end{equation}
\tag{1.8}
Remark 1. According to (1.8), \mathcal{L}(r,v) = r\langle c,dv\rangle_{L^2(T^* X)} for each r \in \mathbb{R}, and hence by definition (1.7) a constant function u=r is a weak solution of equation (1.6) if and only if \theta(r,v)=0 for any function v \in C^\infty(X), where
\begin{equation}
\theta \colon \mathbb{R} \times C^\infty(X) \ni (r,v)\mapsto r\langle c,dv\rangle_{L^2(T^* X)}-\langle f(\,\cdot\, ,r),v\rangle- \langle h,dv\rangle_{L^2(T^* X)} \in \mathbb{R}.
\end{equation}
\tag{1.9}
It is clear that any value of the functional \theta is the difference between the left- and right-hand sides of equation (1.7) for u=r. Remark 2. For a fixed r, the functional \theta(r,\,\cdot\,) is a linear form on C^\infty(X). The space C^\infty(X) is dense in W^{1,2}(X), and hence both the linear form \theta(r,\,\cdot\,) (see (1.9)) and the bilinear form \mathcal{L} (see (1.8)) can be uniquely extended by continuity to functions v \in W^{1,2}(X). Moreover, if either of \theta(r,v)=0 or (1.7) holds for any function v \in C^\infty(X), then it also holds for all functions v \in W^{1,2}(X). It is clear that the fulfillment of any of these equalities for all v \in W^{1,2}(X) is equivalent to that for only non-negative v \in W^{1,2}(X). 1.3. Parabolic equationsThe operator L (1.3) is elliptic, and hence the evolutionary equation is parabolic. By a weak (or generalized) solution of equation (1.10) on the half-open interval [0,T), T \in (0,+\infty], with initial value we mean a function q \in W_{\mathrm{loc}}(T;X) such that f(\,\cdot\, ,q) \in L_{\mathrm{loc}}^2([0,T) \times X), and
\begin{equation}
\langle q(t),p(t)\rangle+\mathcal{L}^t(q,p)=\langle q_0,p(0)\rangle+ \int_0^t\bigl(\langle f(\,\cdot\, ,q(\tau)),p(\tau)\rangle+ \langle h,dp(\tau)\rangle_{L^2 (T^* X)}\bigr)\,d\tau
\end{equation}
\tag{1.12}
for each p \in C^\infty([0,T) \times X) and t \in [0,T), where
\begin{equation}
\mathcal{L}^t \colon W(T;X) \times C^\infty([0,T)\times X) \ni (q,p) \mapsto \int_0^t\bigl(\mathcal{L}(q(\tau),p(\tau))- \langle q(\tau),p'(\tau)\rangle\bigr)\,d\tau \in \mathbb{R}.
\end{equation}
\tag{1.13}
Remark 3. By definitions (1.7) and (1.12), if the initial value q_0 (see (1.11)) is a solution of equation (1.6), then q=q_0 is a solution of the Cauchy problem (1.10), (1.11) on the infinite interval [0,+\infty). Hence, in the case of a constant initial value q_0=r \in \mathbb{R}, according to Remark 1, the constant function q=r is a weak solution of the Cauchy problem (1.10), (1.11) if and only if \theta(r,v)=0 for any function v \in C^\infty(X). Remark 4. The space C^\infty([0,T) \times X) is dense in W^{1,2}([0,T) \times X), and hence, as in Remark 2, the bilinear form \mathcal{L}^t (1.13) can be uniquely extended by continuity to functions p \in W^{1,2}([0,T) \times X) with traces p(0),p(t) \in L^2(X). In addition, the fulfillment of (1.12) for p \in W^{1,2}([0,T) \times X) is equivalent to that for only non-negative p \in W^{1,2}([0,T) \times X). § 2. Main results2.1. Existence, uniqueness, and regularityLet us study conditions under which a weak solution of the Cauchy problem (1.10), (1.11) exists and is unique. A function u \in W^{1,2}(X) is a weak (or generalized) subsolution (supersolution) of equation (1.6) if f(\,\cdot\, ,u) \in L^2(X) and if
\begin{equation*}
\mathcal{L}(u,v) \leqslant \langle f(\,\cdot\, ,u),v\rangle+ \langle h,dv\rangle_{L^2(T^* X)}\qquad \bigl(\mathcal{L}(u,v)\geqslant \langle f(\,\cdot\, ,u),v\rangle+ \langle h,dv\rangle_{L^2(T^* X)}\bigr)
\end{equation*}
\notag
for any non-negative function v \in C^\infty(X). A function q \in W_{\mathrm{loc}}(T;X) is a weak (or generalized) subsolution (supersolution) of the Cauchy problem (1.10), (1.11) on the half-open interval [0,T), T \in (0,+\infty], if f(\,\cdot\, ,q)\in L_{\mathrm{loc}}^2([0,T) \times X) and, if, for any non-negative function p \in C^\infty([0,T) \times X),
\begin{equation}
\begin{aligned} \, &\langle q(t),p(t)\rangle+\mathcal{L}^t(q,p) \\ &\quad\leqslant \langle q_0,p(0)\rangle \,{+}\int_0^t\bigl(\langle f(\,\cdot\, ,q(\tau)),p(\tau)\rangle \,{+}\,\langle h,dp(\tau)\rangle_{L^2(T^* X)}\bigr)\,d\tau \\ &\biggl(\langle q(t),p(t)\rangle+ \mathcal{L}^t(q,p) \\ &\quad\geqslant\langle q_0,p(0)\rangle \,{+}\int_0^t\bigl(\langle f(\,\cdot\, ,q(\tau)),p(\tau)\rangle \,{+}\,\langle h,dp(\tau)\rangle_{L^2(T^* X)}\bigr)\,d\tau\biggr),\qquad t \in [0,T). \end{aligned}
\end{equation}
\tag{2.1}
A weak subsolution (supersolution) of problem (1.10), (1.11) which is not a weak solution will be called strong. In what follows, unless otherwise explicitly stipulated, we will assume that all solutions, subsolutions, and supersolutions are weak, and the adjective “weak” will be dropped for brevity. Remark 5. It is clear that each solution is simultaneously a subsolution and a supersolution. Conversely, if a function is simultaneously a subsolution and a supersolution, then by the concluding lines of Remarks 2 and 4, this function is a solution. It is easily checked that if w \in W^{1,2}(X) is a subsolution (supersolution) of equation (1.6) and w \leqslant q_0 (w \geqslant q_0) almost everywhere, then q=w is a subsolution (supersolution) of problem (1.10), (1.11) on the infinite interval [0,+\infty). Next, in analogy with Remark 1, the constant function u=r, r \in \mathbb{R}, is a subsolution (supersolution) of equation (1.6) if and only if \theta(r,v) \leqslant 0 (\theta(r,v) \geqslant 0) for each non-negative function v \in C^\infty(X). Correspondingly, in analogy with Remark 3, if r \leqslant q_0 (r \geqslant q_0) almost everywhere and \theta(r,v) \leqslant 0 (\theta(r,v) \geqslant 0) for each non-negative function v \in C^\infty(X), then q=r is a subsolution (supersolution) of problem (1.10), (1.11). The following result holds. Theorem 1 (existence and uniqueness of a solution). Assume that a metric a\in L^\infty\bigl((T^* X)^{\otimes^2}\bigr) satisfies estimate (1.2), a vector field b lies in L^\infty(TX), differential forms c,h lie in L^\infty(T^* X), there exists a number \mu \in \mathbb{R} such that
\begin{equation}
\langle c,dv\rangle_{L^2(T^* X)}+\langle \mu,v\rangle \geqslant 0
\end{equation}
\tag{2.2}
for any non-negative function v \in C^\infty(X), and f \in L_{\mathrm{loc}}^\infty(X \times \mathbb{R}) satisfies the Lipschitz condition (see (1.5)) almost everywhere. Let q_1 and q_2 be, respectively, a subsolution and supersolution of problem (1.10), (1.11) on [0,T), T \in (0,+\infty], q_1,q_2 \in L^\infty([0,T) \times X). Then on [0,T) there exists a unique solution q of this problem, and q_1(t) \leqslant q(t) \leqslant q_2(t) almost everywhere for t \in [0,T). For a proof, see § 3. It is known that solutions of the Cauchy problem (1.10), (1.11) are locally Hölder-continuous. More precisely, the following result holds. Theorem 2 (regularity of the solution). Let the coefficients and right-hand side of (1.10) satisfy all the conditions of Theorem 1 except (2.2). Let q be a solution of the Cauchy problem (1.10), (1.11) from L^\infty([0,T) \times X), T \in (0,+\infty). Then q \in C(T;L^2(X)) and, for each (t,x) \in (0,T] \times X, there exist its neighbourhood U and a number 0<\alpha<1 such that q_{|U\cap [0,T) \times X}\in C^{0,\alpha}(U \cap [0,T)\times X). Under the above constraints on the coefficients of the differential operator L (see (1.3)) and the function f (see (1.4)), the result of Theorem 2 follows from the known properties of solutions of linear parabolic equations (see Chap. VI, § 7 in [16] and § 1.5 in [17]). It is natural that a further increase of regularity of the coefficients of equation (1.10) increases correspondingly the regularity of its solutions (see Chap. VI, § 2 in [16]). The following result is clear from Theorems 1 and 2. Theorem 3 (global existence and uniqueness). Let the coefficients and right-hand side of equation (1.10) satisfy the conditions of Theorem 1. If q_1 and q_2 are, respectively, a subsolution and a supersolution of problem (1.10), (1.11) on [0,T), T \in (0,+\infty], and q_1,q_2 \in L_{\mathrm{loc}}^\infty ([0,T) \times X), then on [0,T) problem (1.10), (1.11) has a unique solution q, which is locally Hölder-continuous on (0,T) \times X, q \in C(T;L^2(X)), and q_1(t) \leqslant q(t) \leqslant q_2(t) almost everywhere for t \in [0,T). 2.2. Stabilization of solutionsLet us find conditions under which the solution q of the Cauchy problem (1.10), (1.11) tends as t\to +\infty to the solution u of the stationary equation (1.6). A function f (see (1.4)) is called strictly right (left) concave at a point r_0 \in \mathbb{R} if
\begin{equation}
(1-\alpha)f(\,\cdot\, ,r_0)+\alpha f(\,\cdot\, ,r)< f\bigl(\,\cdot\, ,(1-\alpha) r_0+\alpha r\bigr)
\end{equation}
\tag{2.3}
almost everywhere for r>r_0 (r<r_0) and 0<\alpha<1. Similarly, a function f is called strictly right (left) convex at a point r_0 \in \mathbb{R} if
\begin{equation}
f\bigl(\,\cdot\, ,(1-\alpha)r_0+\alpha r\bigr)<(1-\alpha)f(\,\cdot\, ,r_0)+ \alpha f(\,\cdot\, ,r)
\end{equation}
\tag{2.4}
almost everywhere for r>r_0 (r<r_0) and 0<\alpha<1. The solutions of problem (1.10), (1.11) have the following asymptotic properties. Theorem 4 (stabilization to a stationary solution). Let the coefficients and right-hand side of equation (1.10) satisfy the conditions of Theorem 1, let r_0,N \in \mathbb{R}, r_0 \ne N, and let functions q_0 \in L^2(X) and w \in W^{1,2}(X)\cap L^\infty(X) be such that w \geqslant 0 almost everywhere, V(x \in X\mid w \ne 0)>0 and V(x \in X \mid q_0 \ne r_0)>0. Then the following hold. (a) Let f be strictly right concave at r_0, let r_0 and N be, respectively, a subsolution and supersolution of equation (1.6), let the initial value (1.11) satisfy r_0 \leqslant q_0 \leqslant N almost everywhere, and let there exist \delta such that r_0+\alpha w is a subsolution of equation (1.6) for 0<\alpha<\delta. Then both equation (1.6) and the Cauchy problem (1.10), (1.11) have unique solutions u and q, which are continuous, r_0< u, q(t) \leqslant N for 0<t<+\infty, and satisfy the limit relation (b) Let f be strictly left convex at r_0, let N and r_0 be, respectively, a subsolution and supersolution of equation (1.6), let the initial value (1.11) satisfy N \leqslant q_0 \leqslant r_0 almost everywhere, and let there exist \delta such that r_0-\alpha w is a supersolution of equation (1.6) for 0<\alpha<\delta. Then both equation (1.6) and the Cauchy problem (1.10), (1.11) have unique solutions u and q, which are continuous, N \leqslant u,q(t)<r_0 for 0<t<+\infty, and satisfy (2.5). For a proof, see § 4. Remark 6. The idea of the proof is to construct a non-increasing (in t) function \widetilde{q} \in C([0,+\infty)\times X) such that Theorem 5 (stabilization to a constant solution). Let the coefficients and right-hand side of equation (1.10) satisfy the conditions of Theorem 1, let r_0,A \in \mathbb{R}, A>0, let q_0 \in L^2(X), w \in W^{1,2}(X) \cap L^\infty(X), and let w \geqslant 0 almost everywhere Then the following hold. (a) Let the initial value (1.11) satisfy r_0 \leqslant q_0 \leqslant r_0+Aw almost everywhere, let r_0 be a solution and r_0+\alpha w be a strong supersolution of equation (1.6) for 0<\alpha \leqslant A. Then the Cauchy problem (1.10), (1.11) has a unique solution q, which is continuous, r_0 \leqslant q \leqslant r_0+Aw almost everywhere for 0<t<+\infty, and a unique solution u of equation (1.6) satisfying r_0 \leqslant u \leqslant r_0+Aw almost everywhere is u=r_0, and the limit relation holds (b) Let the initial value (1.11) satisfy r_0-Aw \leqslant q_0 \leqslant r_0 almost everywhere, let r_0 be a solution and r_0-\alpha w be a strong subsolution of equation (1.6) for 0<\alpha \leqslant A. Then the Cauchy problem (1.10), (1.11) has a unique solution q, which is continuous, r_0-Aw \leqslant q \leqslant r_0 almost everywhere for 0<t<+\infty, and a unique solution u of equation (1.6) satisfying r_0-Aw \leqslant u \leqslant r_0 almost everywhere is u=r_0, and the limit relation (2.7) holds. For a proof, see § 4; moreover, Remark 6 with u=r_0 applies as in Theorem 4. Remark 7. Under the conditions of Theorems 4 and 5, using standard a priori estimates of solutions of linear second-order parabolic equations (see Chap. VI, § 1 in [16] and § 1.5 in [17]) and applying Theorems 4 and 5, it can be shown that the solutions of problem (1.10), (1.11) are asymptotically stable. 2.3. The eigenvalue problemAssume that the function f (see (1.4)) has the right (left) derivative
\begin{equation*}
\begin{gathered} \, \frac{\partial f}{\partial r}(\,\cdot\, ,r_0+0)=\lim_{r \to r_0+0} \frac{f(\,\cdot\, ,r)-f(\,\cdot\, ,r_0)}{r-r_0} \\ \biggl(\frac{\partial f}{\partial r}(\,\cdot\, ,r_0-0)= \lim_{r\to r_0-0}\frac{f(\,\cdot\, ,r)-f(\,\cdot\, ,r_0)}{r-r_0}\biggr), \end{gathered}
\end{equation*}
\notag
almost everywhere at a point r_0 \in \mathbb{R}. For the differential operator L (see (1.3)), consider the eigenvalue problem with the potential
\begin{equation}
F(x)=\frac{\partial f}{\partial r}(x,r_0+0)\qquad \biggl(F(x)=\frac{\partial f}{\partial r}(x,r_0-0)\biggr).
\end{equation}
\tag{2.9}
We will say that w \in W^{1,2}(X) is an eigenfunction belonging to an eigenvalue \lambda \in \mathbb{R} if it satisfies equality (2.8) in the weak sense, that is, for all functions v \in C^\infty(X). The following result is known. Theorem 6. Let the coefficients of the operator L (see (1.3)) satisfy the conditions of Theorem 1. If F \in L^\infty(X), then problem (2.8)–(2.10) has a unique simple eigenvalue \lambda_1 to which there belongs a real eigenfunction w \in C^{0,\alpha}(X), 0<\alpha<1, which does not vanish on X. The required result follows from the Krein–Rutman theorem (see § 6.5.2 in [18], § 6.4 in [19], Chap. 11, § C in [20], and Chap. I, § 7 in [21]) and from the regularity of solutions of elliptic equations (see § 8.9 in [22]). The eigenvalue \lambda_1 is known as the principal eigenvalue. From definition (2.3) it easily follows that the function f (see (1.4)) is strictly right (left) concave at a point r_0 \in \mathbb{R} if and only if
\begin{equation}
\frac{f(\,\cdot\, ,r_1)-f(\,\cdot\, ,r_0)}{r_1-r_0}< \frac{f(\,\cdot\, ,r)-f(\,\cdot\, ,r_0)}{r-r_0}
\end{equation}
\tag{2.11}
almost everywhere for r_0<r<r_1 (r_1<r<r_0). Similarly, from definition (2.4) it easily follows that a function f is strictly right (left) convex at a point r_0 \in \mathbb{R} if and only if
\begin{equation}
\frac{f(\,\cdot\, ,r_1)-f(\,\cdot\, ,r_0)}{r_1-r_0}> \frac{f(\,\cdot\, ,r)-f(\,\cdot\, ,r_0)}{r-r_0}
\end{equation}
\tag{2.12}
almost everywhere for r_0<r<r_1 (r_1<r<r_0). The monotonicity properties of the difference relations (2.11) and (2.12) imply that the derivatives in the right-hand sides of (2.9) exist and lie in L^\infty(X). As a corollary, problem (2.8)–(2.10) is well posed. The following results hold. Theorem 7. Let the coefficients and right-hand side of equation (1.10) satisfy the conditions of Theorem 1, and let \lambda_1<0 be the principal eigenvalue. Then the following hold. (a) Let f be strictly right concave at r_0 \in \mathbb{R}, let r_0 and N, r_0<N, be, respectively, a subsolution and supersolution of equation (1.6), and let the initial value (1.11) satisfy r_0\leqslant q_0\leqslant N almost everywhere and V(x\in X\mid q_0 \ne r_0)>0. Then both equation (1.6) and the Cauchy problem (1.10), (1.11) have unique solutions u and q, which are continuous, r_0<u,q(t)\leqslant N for 0<t<+\infty, and satisfy the limit relation (2.5). (b) Let f be strictly left convex at r_0 \in \mathbb{R}, let r_0 and N, N<r_0, be, respectively, a supersolution and subsolution of equation (1.6), and let the initial value (1.11) satisfy N\leqslant q_0\leqslant r_0 almost everywhere and V(x\in X\mid q_0 \ne r_0)>0. Then both equation (1.6) and the Cauchy problem (1.10), (1.11) have unique solutions u and q, which are continuous, N \leqslant u,q(t)<r_0 for 0<t\leqslant +\infty, and satisfy the limit relation (2.5). For a proof, see § 5; moreover, Remark 6 applies as in Theorem 4. Theorem 8. Let the coefficients and right-hand side of equation (1.10) satisfy the conditions of Theorem 1, let a number r_0 \in \mathbb{R} be a solution of equation (1.6), and let q_0 \in L^\infty(X) and \lambda_1 \geqslant0. Then the following hold. (a) Let f be strictly right concave at r_0 and the initial value (1.11) satisfy the inequality q_0 \geqslant r_0 almost everywhere. Then the Cauchy problem (1.10), (1.11) has a unique solution q, which is continuous, r_0 \leqslant q(t) for 0<t\leqslant +\infty, and equation (1.6) has a unique solution u such that r_0 \leqslant u almost everywhere, u=r_0, and which satisfies (2.7). (b) Let f be strictly left concave at r_0 and let the initial value (1.11) satisfy q_0 \leqslant r_0 almost everywhere. Then the Cauchy problem (1.10), (1.11) has a unique solution q, which is continuous, satisfies q(t)\leqslant r_0 for 0<t\leqslant +\infty, and equation (1.6) has a unique solution u such that u\leqslant r_0 almost everywhere, u=r_0, and which satisfies (2.7). For a proof, see § 5. In addition, Remark 6 with u=r_0 applies as in Theorem 4. Moreover, similarly to Remark 7, under the conditions of Theorems 7 and 8, the solutions of problem (1.10), (1.11) are asymptotically stable. § 3. Existence and uniqueness of solutions3.1. Auxiliary resultsFor a proof of Theorem 1, we will require two technical results. The first one is the following comparison test. Lemma 1. Let the coefficients of the differential operator L (see (1.3)) and the function f (see (1.4)) satisfy the conditions of Theorem 1. If q_1,q_2\in W(T;X)\cap L^\infty([0,T) \times X), T\in (0,+\infty], and non-negative function p\in C^\infty([0,T) \times X) satisfy
\begin{equation*}
\begin{aligned} \, &\langle q_1(t),p(t)\rangle+\mathcal{L}^t(q_1,p)-\langle q_1(0),p(0)\rangle- \int_0^t\langle f(\,\cdot\, ,q_1(\tau)),p(\tau)\rangle \,d\tau \\ &\qquad\leqslant \langle q_2(t),p(t)\rangle+\mathcal{L}^t(q_2,p)- \langle q_2(0),p(0)\rangle- \int_0^t\langle f(\,\cdot\, ,q_2(\tau)),p(\tau)\rangle\,d\tau \end{aligned}
\end{equation*}
\notag
for t\in[0,T), and q_1 (0)\leqslant q_2 (0) almost everywhere, then q_1 (t)\leqslant q_2 (t) almost everywhere for t\in[0,T). In addition, if V\bigl(\{x \in X\mid q_1(0) \ne q_2(0)\}\bigr)>0, then q_1(t)<q_2(t) almost everywhere for t\in(0,T). Proof. By the Lipschitz condition (1.5), if
\begin{equation*}
\mu_0=\mu_0(\max\{\|q_1\|_{L^\infty([0,T) \times X)}, \|q_2\|_{L^\infty([0,T) \times X)}\})
\end{equation*}
\notag
then
\begin{equation*}
f(\,\cdot\, ,q_1(\tau))-f(\,\cdot\, ,q_2(\tau))\leqslant \mu_0\bigl(q_1(\tau)-q_2(\tau)\bigr) \operatorname{sgn}\bigl(q_1(\tau)-q_2(\tau)\bigr)
\end{equation*}
\notag
almost everywhere. Hence, by the assumption, for q=q_1-q_2 and for a non-negative p\in C^\infty([0,T) \times X), we have
\begin{equation*}
\langle q(t),p(t)\rangle+\mathcal{L}^t(q,p)-\langle q(0),p(0)\rangle- \int_0^t\langle \mu_0 q(\tau)\operatorname{sgn}q(\tau), p(\tau)\rangle\,d\tau \leqslant 0.
\end{equation*}
\notag
By definition of \mathcal{L} (see (1.8)), \mathcal{L}(e^{\mu t}q,p)=\mathcal{L}(q,e^{\mu t}p). Hence, by definition of \mathcal{L}^t (see (1.13)) and substituting q=e^{\mu t}\widetilde{q} and \widetilde{p}=e^{\mu t}p, the last inequality assumes the form
\begin{equation*}
\langle \widetilde{q}(t),\widetilde{p}(t)\rangle+ \mathcal{L}^t(\widetilde{q},\widetilde{p})- \langle q(0),\widetilde{p}(0)\rangle+\int_0^t \langle(\mu-\mu_0\operatorname{sgn}q(\tau)) \widetilde{q}(\tau),\widetilde{p}(\tau)\rangle\,d\tau \leqslant0
\end{equation*}
\notag
with non-negative \widetilde{p}\in C^\infty([0,T) \times X). For sufficiently large \mu>0, the conclusion of the lemma follows from inequality (2.2) and the strong parabolic maximum principle (see Chap. VI, § 7 in [16] and § 1.5 in [17]). Lemma is proved. Note that that conclusion of Lemma 1 is true, in particular, if q_1(0)=q_2(0) almost everywhere, and also if f \equiv 0. The second technical result we require is the following a priori estimate. Lemma 2. Let the coefficients of the differential operator L (see (1.3)) satisfy the conditions of Theorem 1. If q_0 \in L^2(X), q,v \in W(T;X), T\in (0,+\infty], and p\in C^\infty([0,T) \times X) satisfy
\begin{equation*}
\langle q(t),p(t)\rangle+\mathcal{L}^t(q,p)=\langle q_0,p(0)\rangle+ \int_0^t\langle v(\tau),p(\tau)\rangle\,d\tau,\qquad t\in[0,T),
\end{equation*}
\notag
then there exists a constant C>0, independent of q_0 and v, such that A proof of Lemma 2 follows from the known a priori estimates for solutions of linear second-order parabolic equations (see Chap. VI, § 1 in [16] and § 1.5 in [17]). Note that, in particular, Lemma 2 holds with q_0 \equiv 0. 3.2. Proof of Theorem 1By the assumption of the theorem and definition (2.1),
\begin{equation*}
\begin{aligned} \, &\langle q_1(t),p(t)\rangle+\mathcal{L}^t(q_1,p)-\langle q_0,p(0)\rangle- \int_0^t\langle f(\,\cdot\, ,q_1(\tau)),p(\tau)\rangle\,d\tau \\ &\qquad\leqslant \langle q_2(t),p(t)\rangle+ \mathcal{L}^t(q_2,p)-\langle q_0,p(0)\rangle- \int_0^t\langle f(\,\cdot\, ,q_2(\tau)),p(\tau)\rangle\,d\tau \end{aligned}
\end{equation*}
\notag
for non-negative p \in C^\infty([0,T) \times X) and t \in [0,T). By Lemma 1, we have q_1(t) \leqslant q_2(t) almost everywhere for t\in[0,T). The function f (see (1.4)) satisfies condition (1.5) almost everywhere. Hence, if
\begin{equation*}
\mu \geqslant \mu_0(\max\{\|q_1\|_{L^\infty([0,T) \times X)}, \|q_2\|_{L^\infty([0,T) \times X)}\})
\end{equation*}
\notag
and if functions v_1 and v_2 are such that q_1\leqslant v_1 \leqslant v_2 \leqslant q_2 almost everywhere, then
\begin{equation*}
f(\,\cdot\, ,v_1)+\mu v_1 \leqslant f(\,\cdot\, ,v_2)+\mu v_2
\end{equation*}
\notag
almost everywhere (cf. the appendix to Chap. IV, § 2 in [23] and [24]), and, therefore,
\begin{equation}
\langle f(\,\cdot\, ,v_1)+\mu v_1,v\rangle \leqslant \langle f(\,\cdot\, ,v_2)+\mu v_2,v\rangle
\end{equation}
\tag{3.1}
for any non-negative function v \in C^\infty(X). It is clear that (1.10) is equivalent to the equation
\begin{equation*}
\frac{\partial q}{\partial t}+Lq+\mu q=f(\,\cdot\, ,q)+d_g^* h+\mu q.
\end{equation*}
\notag
Let us construct successive approximations \{q_{1,l}\} and \{q_{2,l}\} by induction. We set q_{1,0}=q_1, q_{2,0}=q_2, and as q_{j,l} for j=1,2 and l=1,2,\dots we take the solution of the Cauchy problem
\begin{equation*}
\frac{\partial q_{j,l}}{\partial t}+Lq_{j,l}+\mu q_{j,l}= f(\,\cdot\, ,q_{j,l-1})+d_g^* h+\mu q_{j,l-1},\qquad q_{j,l}(0)=q_0,
\end{equation*}
\notag
l=1,2,\dots, that is, by definition (1.12), for any function p\in C^\infty([0,T) \times X) and t\in [0,T), we have
\begin{equation}
\begin{aligned} \, &\langle q_{j,l}(t),p(t)\rangle+\mathcal{L}^t(q_{j,l},p)+ \mu\int_0^t\langle q_{j,l}(\tau),p(\tau)\rangle\,d\tau \nonumber \\ &\ =\langle q_0,p(0)\rangle+\int_0^t \bigl(\langle f(\,\cdot\, ,q_{j,l-1}(\tau))+\mu q_{j,l-1}(\tau),p(\tau)\rangle+ \langle h,dp(\tau)\rangle_{L^2(T^* X)}\bigr)\,d\tau. \end{aligned}
\end{equation}
\tag{3.2}
The linear equation (3.2) has a unique solution q_{j,l}\in W(T;X), see Chap. VI, § 1 in [16] and § 1.5 in [17]. Hence, by monotonicity (3.1) and by construction (3.2),
\begin{equation*}
\begin{aligned} \, &\langle q_{j,l}(t),p(t)\rangle+\mathcal{L}^t(q_{j,l},p)+ \mu\int_0^t\langle q_{j,l}(\tau),p(\tau)\rangle\,d\tau \\ &\qquad\leqslant \langle q_{j,l-1}(t),p(t)\rangle+ \mathcal{L}^t(q_{j,l-1},p)+ \mu\int_0^t\langle q_{j,l-1}(\tau),p(\tau)\rangle\,d\tau. \end{aligned}
\end{equation*}
\notag
Hence by Lemma 1, for t\in[0,T),
\begin{equation}
q_1(t)\leqslant q_{1,1}(t) \leqslant q_{1,2}(t)\leqslant\dots\leqslant q_{2,2}(t) \leqslant q_{2,1}(t) \leqslant q_2(t)
\end{equation}
\tag{3.3}
almost everywhere. By monotonicity and since \{q_{1,l}\} and \{q_{2,l}\} are bounded, the limits
\begin{equation}
q_{1,\infty}=\lim_{l \to +\infty}q_{1,l},\qquad q_{2,\infty}=\lim_{l\to +\infty}q_{2,l}
\end{equation}
\tag{3.4}
exist almost everywhere and satisfy
\begin{equation}
q_1(t) \leqslant q_{1,1}(t) \leqslant q_{1,2}(t)\leqslant\dots\leqslant q_{1,\infty}(t) \leqslant q_{2,\infty}(t)\leqslant\dots\leqslant q_{2,2}(t) \leqslant q_{2,1}(t) \leqslant q_2(t)
\end{equation}
\tag{3.5}
almost everywhere. Hence by the Lebesgue dominated convergence theorem, the limit relations (3.4) also hold in the norm \|\,{\cdot}\,\|_{L^2([0,t)\times X)} for t\in [0,T). Next, by construction (3.2)
\begin{equation*}
\begin{aligned} \, &\langle q_{j,l+m}(t)-q_{j,l}(t),p(t)\rangle+\mathcal{L}^t(q_{j,l+m}- q_{j,l},p)+\mu\int_0^t\langle q_{j,l+m}(\tau)-q_{j,l}(\tau),p(\tau)\rangle\,d\tau \\ &\quad =\int_0^t\langle f(\,\cdot\, ,q_{j,l-1+m}(\tau))- f(\,\cdot\, ,q_{j,l-1}(\tau))+\mu(q_{j,l-1+m}(\tau) -q_{j,l-1}(\tau)),p(\tau)\rangle\,d\tau, \end{aligned}
\end{equation*}
\notag
and hence by Lemma 2 there exists C>0 such that, for j=1,2 and l,m=1,2,\dots,
\begin{equation*}
\begin{aligned} \, \|q_{j,l+m}-q_{j,l}\|_{W(t;X)} &\leqslant Ce^{Ct}\|f(\,\cdot\, ,q_{j,l-1+m})- f(\,\cdot\, ,q_{j,l-1})\|_{L^2([0,t)\times X)} \\ &\qquad+\mu Ce^{Ct}(\|q_{j,l-1+m}-q_{j,l-1}-q_{j,l+m}+ q_{j,l}\|_{L^2([0,t) \times X)}). \end{aligned}
\end{equation*}
\notag
By the already established \|\,{\cdot}\,\|_{L^2([0,t)\times X)}-convergence of the approximations \{q_{1,l}\} and \{q_{2,l}\} and also from (3.3) and (1.5), it follows that \{q_{1,l}\} and \{q_{2,l}\} are Cauchy sequences also in the norm \|\,{\cdot}\,\|_{W(t;X)} (see (1.2)). Since W(t;X) is complete, these sequences converge to the functions q_{1,\infty} and q_{2,\infty} (see (3.4)). Therefore, by passing to a limit in (3.2), we get
\begin{equation*}
\begin{aligned} \, &\langle q_{j,\infty}(t),p(t)\rangle+L^t(q_{j,\infty},p)= \langle q_0,p(0)\rangle \\ &\qquad\qquad+\int_0^t\bigl(\langle f(\,\cdot\, ,q_{j,\infty}(\tau)),p(\tau)\rangle+ \langle h,dp(\tau)\rangle_{L^2(T^* X)}\bigr)\,d\tau \end{aligned}
\end{equation*}
\notag
for j=0,1, p\in C^\infty([0,T) \times X) and t\in[0,T), that is, q_{j,\infty} are solutions of problem (1.10), (1.11), and, by (3.5), q_1(t) \leqslant q_{1,\infty}(t) \leqslant q_{2,\infty}(t) \leqslant q_2(t) almost everywhere. By Lemma 1, any solution q of problem (1.10), (1.11) satisfies q_{1,\infty}=q=q_{2,\infty} almost everywhere on the half-open interval [0,T). Theorem 1 is proved. § 4. Stabilization of solutions4.1. Auxiliary resultIn the proof of Theorems 4 and 5 we will require the following auxiliary result. Lemma 3. Let the coefficients and right-hand side of equation (1.10) satisfy the conditions of Theorem 1, and let w_1 and w_2 be, respectively, a subsolution and supersolution of equation (1.6) such that w_1,w_2 \in L^\infty(X) and w_1 \leqslant w_2 almost everywhere. Then there exist unique solutions q_1,q_2 of the problems
\begin{equation}
\frac{\partial q_1}{\partial t}+Lq_1 =f(\,\cdot\, ,q_1)+d_g^* h, \qquad q_1(0) =w_1,
\end{equation}
\tag{4.1}
\begin{equation}
\frac{\partial q_2}{\partial t}+Lq_2 =f(\,\cdot\, ,q_2)+d_g^* h, \qquad q_2(0) =w_2.
\end{equation}
\tag{4.2}
Moreover, the solutions q_1,q_2 are locally Hölder-continuous on (0,+\infty)\times X, q_1,q_2 \in C(+\infty;L^2(X)), and w_1 \leqslant q_1(t) \leqslant q_2(t) \leqslant w_2 almost everywhere for t\in[0,+\infty). Furthermore, q_1 is non-decreasing, and q_2 is non-increasing in t, and both q_1,q_2 converge uniformly to the continuous solutions q_{1,\infty} and q_{2,\infty} of equation (1.6): So, the solutions q_1, q_{1,\infty}, q_2 and q_{2,\infty} satisfy
\begin{equation*}
\lim_{t \to +\infty}\|q_1(t)-q_{1,\infty}\|_{C(X)}=0,\qquad \lim_{t \to +\infty}\|q_2(t)-q_{2,\infty}\|_{C(X)}=0,
\end{equation*}
\notag
and, for 0 \leqslant t' \leqslant t'', the estimates hold almost everywhere. Proof. By the conditions of the lemma and in view of Remark 5, the functions w_1 and w_2 are, respectively, a subsolution and a supersolution of problem (1.10), (1.11). Therefore, by Theorem 3, on [0,+\infty), there exist unique solutions q_1 and q_2 of the Cauchy problems (4.1) and (4.2). Moreover, q_1,q_2 \in C(+\infty;L^2(X)), q_1,q_2 are locally Hölder-continuous on (0,+\infty)\times X, and w_1 \leqslant q_1(t) \leqslant q_2(t) \leqslant w_2 almost everywhere for t\in[0,+\infty).
By Lemma 1, w_1 \leqslant q_1(t') \leqslant q_1(t''), and q_2(t'') \leqslant q_2(t') \leqslant w_2 almost everywhere for 0 \leqslant t' \leqslant t''. Since q_1, q_2 are pointwise monotone, the limit functions in (4.3) and (4.4) exist on the manifold X. By construction, q_1, q_{1,\infty}, q_2 and q_{2,\infty} obey (4.5). In view of (4.5) and by the Lebesgue dominated convergence theorem, equalities (4.3) and (4.4) also hold in the norm \|\,{\cdot}\,\|_{L^2(X)}. The coefficients and right-hand side of equation (1.10) do not depend explicitly on t, and hence, by definitions (1.12) and (1.13), we have
\begin{equation}
\begin{aligned} \, &\langle q_j,p\rangle(t_0+t)-\langle q_j,p\rangle(t_0)+ \int_0^t\bigl(L(q_j,p)(t_0+\tau)- \langle q_j,p'\rangle(t_0+\tau)\bigr)\,d\tau \nonumber \\ &\qquad=\int_0^t\bigl(\langle f(\,\cdot\, ,q_j),p\rangle(t_0+\tau)+ \langle h,dp(t_0+\tau)\rangle_{L^2(T^* X)}\bigr)\,d\tau \end{aligned}
\end{equation}
\tag{4.6}
for t_0 \geqslant 0 and p\in C^\infty([0,+\infty)\times X), and hence, for 0 \leqslant t' \leqslant t'',
\begin{equation*}
\begin{aligned} \, &\langle q_j(t''+t)-q_j(t'+t),p(t)\rangle \\ &\quad\qquad+ \int_0^t\bigl(\mathcal{L}(q_j(t''+\tau)-q_j(t'+\tau),p(\tau)) -\langle q_j(t''+\tau)-q_j(t'+\tau),p'(\tau)\rangle\bigr)\,d\tau \\ &\quad=\langle q_j(t'')-q_j(t'),p(0)\rangle+ \int_0^t\langle f(\,\cdot\, ,q_j)(t''+\tau) -f(\,\cdot\, ,q_j)(t'+\tau),p(\tau)\rangle\,d\tau. \end{aligned}
\end{equation*}
\notag
Therefore, by Lemma 2, there exists a constant C>0 such that, for t \in [0,+\infty),
\begin{equation*}
\begin{aligned} \, &\|q_j(t''+\,\cdot\,)-q_j(t'+\,\cdot\,)\|_{W(t;X)} \\ &\ \ \leqslant Ce^{Ct}\bigl(\|f(\,\cdot\, ,q_j)(t''+\,\cdot\,) -f(\,\cdot\, ,q_j)(t'+\,\cdot\,)\|_{L^2([0,t)\times X)}+ \|q_j(t'')-q_j(t')\|_{L^2(X)}\bigr). \end{aligned}
\end{equation*}
\notag
Hence, by the already established \|\,{\cdot}\,\|_{L^2(X)}-convergence (see (4.3) and (4.4)) and also by (4.6) and (1.5), we conclude that, as t_0 \to +\infty, the functions q_1(t_0+\,\cdot\,) and q_2(t_0+\,\cdot\,) satisfy the Cauchy condition in the norm \|\,{\cdot}\,\|_{W(t;X)}. By completeness of W(t;X), they converge in this space to the functions q_{1,\infty} (see (4.3)) and q_{2,\infty} (see (4.4)), respectively. Choosing p \in W^{1,2}(X) in (4.5) and making t_0\to +\infty, this gives
\begin{equation*}
\int_0^t \mathcal{L}(q_{j,\infty},p)\,d\tau= \int_0^t\bigl(\langle f(\,\cdot\, ,q_{j,\infty}),p\rangle+ \langle h,dp\rangle_{L^2(T^* X)}\bigr)\,d\tau
\end{equation*}
\notag
for j=0,1 and t\in [0,T). The integrands in this equality are independent of \tau, and hence q_{1,\infty} and q_{2,\infty} are solutions of equation (1.6). By well-known regularity properties (see § 8.9 in [22] and § 2.2 in [10]), these solutions are continuous, and so the convergence in (4.4) also holds in the norm \|\,{\cdot}\,\|_{C(X)} by Dini’s test. This proves Lemma 3. 4.2. Proof of Theorem 4(a) By conditions of the theorem and in view of Remark 5, the numbers r_0 and N are, respectively, a subsolution and a supersolution of problem (1.10), (1.11). Therefore, by Theorem 3, on the infinite interval [0,+\infty) there exists a unique solution q of problem (1.10), (1.11); moreover, q \in C(+\infty;L^2(X)), is locally Hölder-continuous on (0,+\infty)\times X, and r_0 \leqslant q(t) \leqslant N for t\in[0,+\infty). By definitions (1.12) and (2.1),
\begin{equation*}
\begin{aligned} \, &\langle r_0,p(t)\rangle+\mathcal{L}^t(r_0,p)-\langle r_0,p(0)\rangle- \int_0^t \langle f(\,\cdot\, ,r_0),p(\tau)\rangle\,d\tau \\ &\qquad \leqslant\langle q(t),p(t)\rangle+\mathcal{L}^t(q,p)- \langle q(0),p(0)\rangle- \int_0^t\langle f(\,\cdot\, ,q(\tau)),p(\tau)\rangle\,d\tau \end{aligned}
\end{equation*}
\notag
for all non-negative p \in C^\infty([0,+\infty)\times X), which implies q(t)>r_0 for t \in (0,+\infty) by Lemma 1. In particular, q(1)>r_0, and hence, by the conditions of the theorem, there exists \delta_0>0 such that r_0+\alpha w is a subsolution of equation (1.6) satisfying r_0+\alpha w \leqslant q(1) almost everywhere for 0<\alpha<\delta_0. By Lemma 3, for w_1=r_0+\alpha w and w_2=N, there exist unique solutions q_1 and q_2 of the Cauchy problems (4.1) and (4.2) on the infinite interval [0,+\infty); moreover, q_1,q_2\in C(+\infty;L^2(X)) and q_1,q_2 are locally Hölder-continuous on (0,+\infty)\times X. The \|\,{\cdot}\,\|_{C(X)}-limits q_{1,\infty} (see (4.3)) and q_{2,\infty} (see (4.4)) exist, satisfy equation (1.6), and obey (4.5) almost everywhere. From (4.5) it follows that q_{1,\infty},q_{2,\infty}>r_0 by the strong maximum principle for elliptic equations (see § 8.7 in [22]), and, therefore, by the uniqueness theorem, q_{1,\infty}=q_{2,\infty} (see § 2.3 in [10]). Moreover, for t \in [0,+\infty) and 0<\alpha<\delta_0, by Lemma 1, we have q_1(t) \leqslant q(t+1) and q(t) \leqslant q_2(t), and hence, for u=q_{1,\infty}=q_{1,\infty} and \widetilde{q}(t)=\max\{u-q_1(t),q_2(t)-u\}, we have (2.6), and, as a corollary, the limit relation (2.5). Assertion (b) is verified similarly. Theorem 4 is proved. 4.3. Proof of Theorem 5(a) By conditions of the theorem and in view of Remark 5, r_0 is a subsolution, and r_0+Aw is a supersolution of problem (1.10), (1.11). Hence an appeal to Theorem 3 and Lemma 3 with w_1=r_0 and w_2=r_0+Aw shows that there exist unique solutions q and q_2 of problems (1.10), (1.11) and (4.2) on the infinite interval [0,+\infty), where q,q_2\in C(+\infty;L^2(X)) and q,q_2 are locally Hölder-continuous on (0,+\infty)\times X. Next, by Lemma 3, the \|\,{\cdot}\,\|_{C(X)}-limit q_{2,\infty} exists (see (4.4)) and is a solution of equation (1.6). Furthermore, by (4.5) r_0 \leqslant q_{2,\infty} \leqslant r_0+Aw almost everywhere. Hence q_{2,\infty}=r_0 by the well-known uniqueness theorem (see § 2.3 in [10]). Next, by Lemma 1, r_0 \leqslant q(t) \leqslant q_2(t) almost everywhere for t\in [0,+\infty), and hence \widetilde{q}(t)=q_2(t)-r_0 obeys (2.6), and, therefore, and the limit relation (2.7). Assertion (b) is proved similarly. Theorem 5 is proved. § 5. The eigenvalue problem5.1. Proof of Theorem 7(a) By Theorem 6, an eigenfunction w \in C(X) belonging to the eigenvalue \lambda_1 can be chosen so as to have \inf_{x \in X}w(x)\,{>}\,0. By the assumption \lambda_1<0, and hence, by definition of the right derivative there exists \delta>0 such that, for all 0<\alpha<\delta,
\begin{equation*}
\frac{\lambda_1}{2}\alpha w<f(\,\cdot\, ,r_0+\alpha w)- f(\,\cdot\, ,r_0)-\alpha w\,\frac{\partial f}{\partial r}(\,\cdot\, ,r_0+0)
\end{equation*}
\notag
almost everywhere. Now from (2.10), (2.9) we have
\begin{equation*}
\begin{aligned} \, 0&=\alpha\mathcal{L}(w,v)-\alpha\biggl\langle\frac{\partial f}{\partial r} (\,\cdot\, ,r_0+0)w,v\biggr\rangle-\lambda_1\alpha\langle w,v\rangle \\ &\geqslant \alpha\mathcal{L}(w,v)-\langle f(\,\cdot\, ,r_0+\alpha w),v\rangle+ \langle f(\,\cdot\, ,r_0),v\rangle+ \frac{\lambda_1}{2}\alpha\langle w,v\rangle-\lambda_1\alpha\langle w,v\rangle \end{aligned}
\end{equation*}
\notag
for non-negative functions v \in C^\infty(X) (cf. § 3.2 in [3]). Therefore,
\begin{equation*}
\begin{aligned} \, &\alpha \mathcal{L}(w,v)+r_0\langle c,dv\rangle_{L^2(T^* X)}- \langle f(\,\cdot\, ,r_0+\alpha w),v\rangle- \langle h,dv\rangle_{L^2(T^* X)} \\ &\qquad\leqslant r_0\langle c,dv\rangle_{L^2(T^* X)}- \langle f(\,\cdot\, ,r_0),v\rangle-\langle h,dv\rangle_{L^2(T^* X)}+ \frac{\lambda_1}{2}\alpha\langle w,v\rangle. \end{aligned}
\end{equation*}
\notag
Hence by the definitions of \mathcal{L} (see (1.8)) and \theta (see (1.9)), for all 0<\alpha<\delta, we have
\begin{equation*}
\mathcal{L}(r_0+\alpha w,v)-\langle f(\,\cdot\, ,r_0+\alpha w),v\rangle- \langle h,dv\rangle_{L^2(T^* X)} \leqslant \theta(r_0,v)+\frac{\lambda_1}{2}\alpha\langle w,v\rangle.
\end{equation*}
\notag
By Remark 5, a number r_0 is a subsolution of (1.6) if and only if \theta(r_0,v) \leqslant 0 for each non-negative function v \in C^\infty(X). Next, since \inf_{x \in X}w(x)>0 and \lambda_1<0, and hence, for 0<\alpha<\delta, the function r_0+\alpha w is a subsolution of equation (1.6). So, the conditions of assertion (a) of Theorem 4 are met. Applying this assertion we get the required result. Assertion (b) is verified similarly. Theorem 7 is proved. 5.2. Proof of Theorem 8(a) By Theorem 6, an eigenfunction w \in C(X) belonging to the eigenvalue \lambda_1 can be chosen so as to have \inf_{x\in X}w(x) > 0. Hence, by monotonicity of the difference relations (2.11), for any \alpha>0, we have
\begin{equation}
f(\,\cdot\, ,r_0+\alpha w)-f(\,\cdot\, ,r_0)< \alpha w\,\frac{\partial f}{\partial r}(\,\cdot\, ,r_0+0)
\end{equation}
\tag{5.1}
almost everywhere. Now an appeal to (2.10), (2.9) shows that
\begin{equation*}
\begin{aligned} \, 0&=\alpha\mathcal{L}(w,v)- \alpha\biggl\langle\frac{\partial f}{\partial r} (\,\cdot\, ,r_0+0)w,v\biggr\rangle-\lambda_1\alpha\langle w,v\rangle \\ &\leqslant \alpha\mathcal{L}(w,v)-\langle f(\,\cdot\, ,r_0+\alpha w),v\rangle+ \langle f(\,\cdot\, ,r_0),v\rangle-\lambda_1\alpha\langle w,v\rangle \end{aligned}
\end{equation*}
\notag
for non-negative functions v \in C^\infty(X) (cf. § 3.2 in [3]). Therefore,
\begin{equation*}
\begin{aligned} \, &\alpha\mathcal{L}(w,v)+r_0\langle c,dv\rangle_{L^2(T^* X)}- \langle f(\,\cdot\, ,r_0+\alpha w),v\rangle-\langle h,dv\rangle_{L^2(T^* X)} \\ &\qquad\geqslant r_0\langle c,dv\rangle_{L^2(T^* X)}- \langle f(\,\cdot\, ,r_0),v\rangle-\langle h,dv\rangle_{L^2(T^* X)}+ \lambda_1 \alpha\langle w,v\rangle, \end{aligned}
\end{equation*}
\notag
and now by definitions of \mathcal{L} (see (1.8)) and \theta (see (1.9)), for any \alpha>0, we have
\begin{equation*}
\mathcal{L}(r_0+\alpha w,v)-\langle f(\,\cdot\, ,r_0+\alpha w),v\rangle- \langle h,dv\rangle_{L^2(T^* X)} \geqslant \theta(r_0,v)+\lambda_1 \alpha\langle w,v\rangle.
\end{equation*}
\notag
In addition, since inequality (5.1) is strict, for v=1 we have
\begin{equation*}
\mathcal{L}(r_0+\alpha w,1)-\langle f(\,\cdot\, ,r_0+\alpha w),1\rangle> \theta(r_0,1)+\lambda_1\alpha\langle w,1\rangle.
\end{equation*}
\notag
By Remark 5, a number r_0 is a solution of equation (1.10) if and only if \theta(r_0,v)=0 for each non-negative function v \in C^\infty(X). Since \inf_{x\in X}w(x)>0 and \lambda_1\geqslant 0, it follows that, for \alpha>0, the function r_0+\alpha w is a strict supersolution of equation (1.6). Let A>0 be such that r_0+Aw \geqslant q_0 almost everywhere. This A satisfies the conditions of assertion (a) of Theorem 5, which, in turn, secures the required result. Assertion (b) is verified similarly. Theorem 8 is proved. The author is grateful to A. A. Davydov for posing the problem and useful discussions.
Citation in format AMSBIB
\Bibitem{Tun23}
This publication is cited in the following 5 articles:
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