We consider a nonlinear one-dimensional scalar equation of diffusion type in which, depending on the gradient of the solution, the diffusion coefficient may be positive or negative. We compare two concepts of Young measure solutions which are based on different methods to construct approximate solutions, the SP-solutions (singular perturbation) and the EM-solution (energy minimization). We show that the SP-solution can recover classical solutions where the EM-solution fails to do so, and that EM-solutions are more stable under perturbations of the initial values. We characterize the EM-solution with a free boundary problem and determine its long-time behavior.

We investigate the continuity of solutions for general nonlinear parabolic equations of the form \begin{equation*} u_t-\sum_{i=1}^n\frac{\partial}{\partial x_i}\Bigl(\Bigl|\frac{\partial u}{\partial x_i}\Bigr|^{p(x)-2}\frac{\partial u}{\partial x_i}\Bigr)=0,\qquad 2 < p_1\leq p(x)\leq p_2 \end{equation*} near a nonsmooth boundary of a cylindrical domain. We prove the sufficient and necessary condition for regularity of a boundary point in terms of the $p(x)$-capacity.

We present existence results for ordinary $p$-Laplacian systems of the form $$ -( | u' | ^{p-2}u')' = f(t,u), \; \; \hbox { in } [0,T], \tag*{(*)}$$ submitted to the general potential boundary condition $$ ((|u'|^{p-2}u')(0), -(|u'|^{p-2}u')(T) ) \in \partial j(u(0), u(T)).$$ Here, $p \in (1, \infty)$ is fixed, $j:\mathbb{R}^N \times \mathbb{R}^N \to (- \infty , +\infty ]$ is proper, convex and lower semicontinuous and $f:[0,T] \times \mathbb{R}^N \to \mathbb{R}^N $ is a Carathéodory mapping. Firstly, we deal with the potential case $f(t,u)=\nabla F(t,u)$, with $F:[0,T] \times \mathbb{R} ^N \to \mathbb{R} $ continuously differentiable with respect to the second variable. Secondly, the system will be a nonpotential one. Afterwards, instead of $(*)$ will be the differential inclusions system $$-( | u' | ^{p-2}u')' \in \overline {\partial } F(t,u), \; \; \hbox { in } [0,T], $$ where, this time, $F$ is only locally Lipschitz with respect to the second variable and $\overline {\partial } F(t,x)$ stands for Clarke's generalized gradient of $F(t, \cdot )$ at $x \in \mathbb{R}^N$. Several examples of applications are given.

We investigate the solvability of the Neumann problem (1.1) involving the critical Sobolev nonlinearity with an indefinite weight function and the Hardy potential. We prove that there exists $\lambda^*>0$ such that for $\lambda \in (0,\lambda^*)$, problem (1.1) admits at least two distinct solutions.

In this paper, we discuss the asymptotic behavior of solutions of nonlinear parabolic equations in ${\mathbf{R}}^N$ involving critical Sobolev exponent. For the semilinear and subcritical problem, it is well-known that the solution which intersects with the ``stable set'' must be a global one and the solution which enters the "unstable set"feodory should blow up in finite time. But in the critical case, it is not clear that the same result holds or not. In this paper, we show that the same result holds also in the critical case. The proof of our main result requires the method different from that for the subcritical problem and is based on the direct analysis of $L^\infty$-norm of solutions with the aid of the blow up argument and the concentration compactness type argument.

In this paper we deal with the study of some regularity properties of weak solutions to non-linear, second-order parabolic equations and systems of the type \[ u_{t}-{\operatorname{div}} A(Du)=0 \;,\;\;\; (x,t)\in \Omega \times (-T,0)=\Omega_{T}, \] where $\Omega \subset {\mathbb{R}}^{n}$ is a bounded domain, $T>0$, $A:{\mathbb{R}}^{nN}\to {\mathbb{R}}^{N}$ satisfies a $p$-growth condition and $u:\Omega_{T}\to {\mathbb{R}}^{N}$. In particular, we focus our attention on local regularity of the spatial gradient of solutions of problems characterized by weak differentiability and ellipticity assumptions on the vector field $A(z)$. We prove the local Lipschitz continuity of solutions in the scalar case ($N=1$). We extend this result in some vectorial cases under an additional structure condition. Finally, we prove higher integrability and differentiability of the spatial gradient of solutions for general systems.