Solvability of the initial-boundary value problem for the quasilinear heat equation (Q2703988)

The article is devoted to studying the following initial-boundary value problem for the quasilinear heat equation: NEWLINE\[NEWLINE \begin{aligned} &C(T)\frac{\partial T}{\partial t} - \text{div}(L \operatorname {grad}T) = f(t,\overline x),\quad t\in (0,T),\quad \overline x\in\Omega,\\ &T|_{t=0} = f_0(\overline x),\quad T|_{\overline x\in\gamma_1}=0,\quad \partial T/\partial\vec n|_{\overline x\in\gamma_2} = 0, \end{aligned} NEWLINE\]NEWLINE where \(\Omega\subset\mathbb R^m\) \((1\leq m\leq 3)\) is a bounded domain with piecewise smooth boundary \(\partial\Omega = \gamma_1\cup\gamma_2\) and \(\overline x = (x_1,\dots,x_m)^T\in\Omega\). The author formulates a generalized statement of the problem weaker than that studied by Ladyzhenskaya and Alt. It is proven the existence and uniqueness of a generalized solution in the class \(L_2((0,T)\times\Omega)\) under the assumption that the thermal conductivity and the heat capacity are bounded, while the source function \(f\) belongs to a wider class than \(L_2\).