## Weak solutions and maximal regularity for abstract evolution inequalities.(English)Zbl 0858.35073

Summary: We study the regularity and the approximation of the solution of a parabolic evolution inequality $u'(t)+A(t)u(t)+ \partial\phi(u(t))\ni f(t), \qquad u(0)=u_0\tag{$$*$$}$ in the framework of a Hilbert triple $$\{V,H,V'\}$$; here $$A(t)$$, $$t\in]0,T[$$, is a family of linear continuous and coercive operators from $$V$$ to $$V'$$ and $$\phi$$ is a proper convex l.s.c. function defined in $$V$$ with values in $$]-\infty,\infty]$$. We give a weak formulation of $$(*)$$ which allows $$f\in H^{-1/2+\delta}(0,T;H)$$, for some $$\delta>0$$ and we prove that the weak solution belongs to $$H^{1/2-\varepsilon} (0,T;H)$$, $$\forall \varepsilon>0$$, besides the standard $$L^2(0,T;V)\cap L^\infty(0,T;H)$$.
Under suitable regularity hypotheses on the data $$f$$ (in particular $$f\in H^{1/2+\delta}(0,T;H)$$ for some $$\delta>0$$ is sufficient) and $$u_0$$, the strong solution is in $$H^{3/2-\varepsilon} (0,T;H)$$, $$\forall \varepsilon>0$$. In this case we prove an optimal error estimate for the backward Euler discretization of $$(*)$$ in $$L^2(0,T;V)\cap L^\infty(0,T;H)$$ and in $$H^1(0,T;H)$$. We apply these results to the porous medium and the Stefan problem.

### MSC:

 35K85 Unilateral problems for linear parabolic equations and variational inequalities with linear parabolic operators 35D10 Regularity of generalized solutions of PDE (MSC2000)