Existence of solutions to a nonlinear parabolic problem with two weights
Volume 152 / 2018
Abstract
The paper is focused on an application of the weighted space $$L^p(0,T; W_{(\omega_1,\omega_2),0}^{1,p}(\Omega))$$ to nonlinear parabolic problems. This framework is involved in studies of parabolic existence in the weighted setting via elliptic existence in the classical setting. We prove that the existence of solutions $v\in W^{1,p}_{\rm loc}(\Omega)$ to an elliptic problem of the form $$-\Delta_p v\geq \Phi(x,v,\nabla v),$$ with $\Phi$ nonnegative or bounded from below in a certain sense, leads to existence of weak solutions $u\in L^p(0,T; W_{(\omega_1,\omega_2),0}^{1,p}(\Omega))$ to a nonlinear parabolic problem $$\begin{cases} u_t-\operatorname{div}(\omega_2(x)|\nabla u|^{p-2} \nabla u )= \lambda \omega_1(x) |u|^{p-2}u,& x\in\Omega,\\ u(x,0)=f(x),& x\in\Omega, \end{cases}$$ where the parameter $\lambda$ is smaller than the optimal constant in the corresponding Hardy inequality, while $f\in L^2(\Omega)$.
The open subset $\Omega\subseteq\mathbb{R}^n$ is not necessarily bounded, and the weight functions $\omega_1,\omega_2$ are not assumed to satisfy the $A_p$-condition.
