Strong full bounded solutions of nonlinear parabolic equations with nonlinear boundary conditions
The authors deal with the following nonlinear parabolic boundary value problem \[ \begin{cases} \frac{\partial u}{\partial t}(x,t)-Lu(x,t)= f(x,t,u) &\text{a.e. in }\Omega \times\mathbb R,\\ {\mathcal B}u=\varphi(x,t,u) &\text{a.e. on }\partial \Omega \times \mathbb R,\\ \sup_{\Omega \times\mathbb R}| u(x,t)| <\infty, \end{cases} \] where \(\Omega \) is a bounded, open and connected subset of \(\mathbb R^N\), \(L\) is a second-order, uniformly elliptic differential operator with time-dependent coefficients, and \({\mathcal B}\) is a linear first-order boundary operator which is of either Dirichlet, Neumann, or regular oblique type. It is supposed that the coefficients of the operators \(L\) and \({\mathcal B}\) are measurable and bounded. The reaction and the boundary nonlinearities \(f\) and \(\varphi \) are Carathéodory functions. The authors are interested in bounded solutions existing for all times. The authors show first, by a counter-example, that the weak maximum principle fails when solutions exits for all times, even when the coefficients in the equation are very smooth. They consider a boundedness condition at minus infinity in order to establish one-sided \(L^\infty\) a priori estimates for the solutions to linear boundary value problems and derive a weak maximum principle which is valid on the entire real line in time. This counter-example shows that such a condition is essential for the maximum or comparison principle to hold. The authors formulate the general assumptions and state the main result concerning the existence, and, in some instance, positivity and uniqueness of bounded solutions existing for all times for nonlinear problems with nonlinear boundary conditions. The existence of minimal or maximal solutions is also discussed requiring only that the reaction nonlinearity \(f\) is Carathéodory, no one-sided Lipschitz nor Hölder continuity are required on \(f\). They prove the main results using some auxiliary results, comparison techniques, Galiardo-Nirenberg type interpolation inequalities, a priori estimates and nonlinear approximation methods. Finally, the authors gives some examples which illustrate the main result.
scientific article; zbMATH DE number 5925810
scientific article