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Differential equations and Optimal control
Initial boundary value problem with nonlocal boundary condition for a nonlinear parabolic equation with memory
A. Gladkov Belarusian State University, 4 Niezaliezhnasci Avenue, Minsk 220030, Belarus
Abstract:
We consider a nonlinear parabolic equation with memory $u_{t}=\Delta u+au^{p}\int\limits_0^t u^{q}(x,\tau)\mathrm{d}\tau-bu^{m}$ for $(x,t)\in \Omega\times (0,+\infty)$ under nonlinear nonlocal boundary condition $\left.\frac{\partial u(x,t)}{\partial v}\right|_{\partial\Omega\times (0,+\infty)}= \int\limits_\Omega k(x,y,t)u^{l}(y,t)\mathrm{d}y$ and initial data $u(x,0)=u_{0}(x), x\in\Omega$,where $a,b,q,m,l$ - are positive constants; $p\geq 0$; $\Omega$ - is a bounded domain in $\mathbb{R}^{n}$ with smooth boundary $\partial\Omega$; $v$ - is unit
outward normal on $\partial\Omega$. Nonnegative continuous function $k(x,y,t)$ is defined for $x\in \partial\Omega, y\in\bar{\Omega}, t\geq 0$, nonnegative function $u_{0}(x)\in C^{1}(\bar\Omega)$, while it satisfies the condition $\frac{\partial u_{0}(x)}{\partial v}=\int\limits_\Omega k(x,y,0)u_{0}^{l}(y)\mathrm{d} y $ for $x\in\partial\Omega$. In this paper we study classical solutions. We establish the existence of a local maximal solution of the original problem. We introduce definitions of a supersolution and a subsolution. It is shown that under some conditions a supersolution is not less than a subsolution. We find conditions for the positiveness of solutions. As a consequence of the positiveness of solutions and the comparison
principle of solutions, we prove the uniqueness theorem.
Keywords:
nonlinear parabolic equation; nonlocal boundary condition; existence of a solution; comparison principle.
Received: 05.05.2023 Revised: 01.06.2023 Accepted: 02.06.2023
Citation:
A. Gladkov, “Initial boundary value problem with nonlocal boundary condition for a nonlinear parabolic equation with memory”, Journal of the Belarusian State University. Mathematics and Informatics, 2 (2023), 18–27
Linking options:
https://www.mathnet.ru/eng/bgumi428 https://www.mathnet.ru/eng/bgumi/v2/p18
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Abstract page: | 70 | Full-text PDF : | 29 | References: | 15 |
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