Wiener criteria for existence of large solutions of nonlinear parabolic equations with absorption in a non-cylindrical domain

TL;DR

This paper establishes Wiener criteria involving parabolic capacity to determine the existence of large solutions for certain nonlinear parabolic equations in non-cylindrical domains, extending classical potential theory results.

Contribution

It introduces new necessary and sufficient Wiener-type conditions for large solutions of nonlinear parabolic equations with absorption, including exponential and gradient terms.

Findings

Wiener criteria characterize existence of large solutions.

Conditions involve parabolic $W_{q'}^{2,1}$-capacity.

Results apply to equations with absorption and gradient terms.

Abstract

We obtain a necessary and a sufficient condition expressed in terms of Wiener type tests involving the parabolic $W\_{q'}^{2,1}$- capacity, where $q'=\frac{q}{q-1}$, for the existence of large solutions to equation $\prt\_tu-\Delta u+u^q=0$ in non-cylindrical domain, where $q\textgreater{}1$. Also, we provide a sufficient condition associated with equation $\prt\_tu-\Delta u+e^u-1=0$ . Besides, we apply our results to equation: $\prt\_tu-\Delta u+a|\nabla u|^p+bu^{q}=0$ for $a,b\textgreater{}0$, $11$.