Investigation of the solvability of a degenerate evolution equation with nonhomogeneous nonlinearity by the compactness method (Q2703983)

In the article under review, the authors study the following problem: find a function \(u(x,t)\) satisfying the equation NEWLINE\[NEWLINE u_t - \Delta \varphi(u(x,t)) = f(x,t),\qquad x\in\Omega,\quad t > 0, NEWLINE\]NEWLINE and the conditions NEWLINE\[NEWLINE u(x,t)|_{\Sigma} = 0,\qquad u(x,0) = u_0(x). NEWLINE\]NEWLINE Here \(\Omega\) is a bounded domain in \(\mathbb R^N\), \(\Gamma = \partial\Omega\), and \(\Sigma = \Gamma\times (0,T)\) is the lateral surface of the cylinder \(Q = \Omega\times (0,T)\), \(T < \infty\). The smoothness of \(\Gamma\) is not assumed. The aim of the article is to prove the existence of a solution to the problem with minimal conditions on the smoothness and other properties of the function \(\varphi\). In particular, the equation can be degenerated at infinite many values of \(u\). The following classes of functions \(\varphi\) are typical examples: functions with a nonpower-law growth and with a degeneration of \(\varphi'(u)\) at infinity; functions whose derivative vanishes at several points; nonpower-law functions with at most power-law growth; nonhomogeneous functions of power-law growth. Under suitable assumptions on the data of the problem, the authors prove the existence of generalized solutions to the problem in a class of unbounded functions.