Stability/nonstability properties of renormalized/entropy solutions for degenerate parabolic equations with $$L^{1}$$/measure data

Abstract

We study the possibility to give a formulation to the degenerate parabolic problems modeled by $$\begin{aligned} (\mathcal {P}_{b}^{1})\ {\left\{ \begin{array}{ll} u_{t}-\text {div}\left[ |\nabla u|^{p-2}\nabla u)\text {/}(1+|u|)^{\theta (p-1)}\right] =\mu \text { in } (0,T)\times \Omega ,\\ u(0,x)=u_{0}(x)\text { in }\Omega ,\quad u(t,x)=0\text { on }(0,T)\times \partial \Omega , \end{array}\right. } \end{aligned}$$ where $$\theta >0$$ , $$u_{0}\in L^{1}(\Omega )$$ and $$\mu $$ is a general (nonnegative) Radon measure. We also investigate the strong stability of solutions for noncoercive absorption problems whose model $$\begin{aligned}(\mathcal {P}_{b}^{2})\ {\left\{ \begin{array}{ll} u_{t}-\text {div}\left[ |\nabla u|^{p-2}\nabla u)\text {/}(1+|u|)^{\theta (p-1)}\right] +|u|^{q-1}u=f\text { in }(0,T)\times \Omega ,\\ u(0,x)=0\text { in }\Omega ,\quad u(t,x)=0\hbox { on }(0,T)\times \partial \Omega \end{array}\right. }\end{aligned}$$ where $$q>r(p-1)[1+\theta (p-1)]\text {/}(r-p)$$ and $$f\in L^{1}_{\text {loc}}(Q\backslash K)$$ with K is a compact subset of Q of zero r-capacity (or, is a measure concentrated on a set of r-capacity zero). We prove the convergence of approximate solutions $$u_{n}$$ (related to a regular approximation $$\mu _{n}$$ of $$\mu $$ ) towards a renormalized solutions u of $$(\mathcal {P}_{b}^{1})$$ , and we extend the previous known-results on the nonstability of entropy solutions for problems $$(\mathcal {P}_{b}^{2})$$ .