Unique solvability of the Cauchy problem for certain quasilinear pseudoparabolic equations

We study the Cauchy problem in the layer $\Pi_T={\mathbb R}^n\times[0,T]$ for the equation
$$ u_t=c\Delta u_t+\Delta\varphi(u), $$
where $c$ is a positive constant and the function $\varphi(p)$ belongs to $C^1({\mathbb R}_+)$ and has a nonnegative monotone non-decreasing derivative. The unique solvability of this Cauchy problem is established for the class of nonnegative functions $u(x,t)\in C_{x,t}^{2,1}(\Pi_T)$ with the properties:
\begin{align*} \varphi'\bigl(u(x,t)\bigr) &\le M_1(1+|x|^2), \\ \|u_t(x,t)| & \le M_2(1+|x|^2)^\beta\qquad (\beta >0). \end{align*}