We report on new results concerning the global well-posedness, dissipativity and attractors for the quintic wave equations in bounded domains of ${ℝ}^3$ with damping terms of the form ${(-{\Delta }_x)}^{\theta }{\partial }_{t}u$, where $\theta =0$ or $\theta =1/2$. The main ingredient of the work is the hidden extra regularity of solutions that does not follow from energy estimates. Due to the extra regularity of solutions existence of a smooth attractor then follows from the smoothing property when $\theta =1/2$. For $\theta =0$ existence of smooth attractors is more complicated and follows...

In this note we give some summability results for entropy solutions of the nonlinear parabolic equation ut - div ap (x, ∇u) = f in ] 0,T [xΩ with initial datum in L1(Ω) and assuming Dirichlet's boundary condition, where ap(.,.) is a Carathéodory function satisfying the classical Leray-Lions hypotheses, f ∈ L1 (]0,T[xΩ) and Ω is a domain in RN. We find spaces of type Lr(0,T;Mq(Ω)) containing the entropy solution and its gradient. We also include some summability results when f = 0 and the p-Laplacian...

We prove boundedness and continuity for solutions to the Dirichlet problem for the equation $$-\mathrm{div}\left(a(x,\nabla u)\right)=h(x,u)+\mu ,\text{in}\Omega \subset {ℝ}^N,$$ where the left-hand side is a Leray-Lions operator from $W_0^{1,p}\left(\Omega \right)$ into $W^{-1,p^{\text{'}}}\left(\Omega \right)$ with $1<p<N$, $h(x,s)$ is a Carathéodory function which grows like ${\left|s\right|}^{p-1}$ and $\mu$ is a finite Radon measure. We prove that renormalized solutions, though not globally bounded, are Hölder-continuous far from the support of $\mu$.

We prove a Sobolev and a Morrey type inequality involving the mean curvature and the tangential gradient with respect to the level sets of the function that appears in the inequalities. Then, as an application, we establish a priori estimates for semistable solutions of $–{\Delta }_{p}u=g\left(u\right)$ in a smooth bounded domain $\Omega \subset {ℝ}^n$. In particular, we obtain new $L^r$ and $W^{1,r}$ bounds for the extremal solution $u^{☆}$ when the domain is strictly convex. More precisely, we prove that $u^{☆}\in L^{\infty }\left(\Omega \right)$ if $n\le p+2$ and $u^{☆}\in L^{\frac{np}{n-p-2}}\left(\Omega \right)\cap W_0^{1,p}\left(\Omega \right)$ if $n>p+2$.