Existence and uniqueness of weak solutions for a multi–phase parabolic equation with L1–weights

Abstract

We study the initial--boundary value problem\[\partial_t u - \divergence\bigl(\Lambda(\nabla u)\bigr) + \Lambda(u) = f(x,u),\quad u|_{\partial\Omega}=0,\]in a bounded domain $\Omega\subset\R^n$ ($n\ge 3$), where\[\Lambda(\tau)=|\tau|^{p-2}\tau + \sum_{i=1}^l \lambda_i(x)|\tau|^{z_i-2}\tau,\]$1<p<z_i<\min\{n,p^*\}$, $z_i<\frac{pn+p}{n}$, and the weights satisfy $0\le \lambda_i\in L^1(\Omega)$.The nonlinearity $f$ is assumed Lipschitz and of linear growth.Using the Galerkin method combined with monotone operator theory inMusielak--Orlicz spaces we prove the global existence and uniquenessof weak solutions. The critical compact embedding\[W^{1,p}_0(\Omega)\hookrightarrow L^{z_i}(\Omega,\lambda_i\dx)\]guaranteed by the structural assumptions is the key ingredientin obtaining a priori estimates and passing to the limit.