The present work proposes a wide survey of the recent results on weak solvability and regularity properties of evolution equations. The authors concentrate their attention on degenerate quasilinear parabolic equations of the type \[ \begin{cases} u\in L^\infty_{\text{loc}} (0,T;W^{1,p}_{\text{loc}}(\Omega)), \quad p>1,\cr u_t-\text{div} {\mathbb A}(x,t,u,\nabla u)=B(x,t,u,\nabla u), \text{ weakly in } \Omega_T=\Omega\times(0,T). \end{cases}(1)_p \] The functions \({\mathbb A}=(A_1,\ldots,A_N)\) and \(B\) are real valued, measurable, and satisfying the structure conditions \[ \begin{cases} C_0| \nabla u| ^{p-2}| \nabla u| ^2-C\leq {\mathbb A}(x,t,u,\nabla u)\cdot\nabla u,\cr | {\mathbb A}(x,t,u,\nabla u)| +| B(x,t,u,\nabla u)| \leq C(1+| \nabla u| ^{p-1}) \end{cases}(2)_p \] where \(C_0\) and \(C\) are given positive constants. The prototype example is \[ u_t-\text{div}| \nabla u| ^{p-2}\nabla u =0\qquad\text{ for some } p>1. (3)_p \] Along with \((1)_p-(2)_p\) the authors consider also the quasilinear equations \[ \begin{cases} u\in L^\infty_{\text{loc}} (0,T;W^{1,2}_{\text{loc}}(\Omega)),\cr u_t-\text{div} {\mathbb A}(x,t,u,\nabla u)=B(x,t,u,\nabla u), & \text{ weakly in } \Omega_T=\Omega\times(0,T). \end{cases}(1)_m \] with structure conditions \[ \begin{cases} C_0 | u| ^{m-1}| \nabla u| ^2-C\leq {\mathbb A}(x,t,u,\nabla u)\cdot\nabla u,\cr | {\mathbb A}(x,t,u,\nabla u)| +| B(x,t,u,\nabla u)| \leq C| u| ^{m-1}(1+| \nabla u| ), & m>0, \end{cases}(2)_m \] and the prototype example is \[ u_t-\Delta | u| ^{m-1} u =0\qquad\text{ for some } m>0. (3)_m \] The authors also consider singular equations of the Stefan-type where the singularity occurs in the time-part of the operator \[ \begin{cases} u\in C_{\text{loc}}(0,T; W^{1,2}_{\text{loc}}(\Omega_T)),\cr \beta(u)_t-\text{div}{\mathbb A}(x,t,u,\nabla u)\ni B(x,t,u,\nabla u) & \text{ in } D'(\Omega_T), \end{cases} (1)_S \] where \(\mathbb A\) and \(B\) have the same structure conditions as \((2)_m\) for \(m=1\) and \(\beta (\cdot)\) is a coercive, maximal monotone graph in \({\mathbb R}\times {\mathbb R}.\) The prototype example is \[ \beta(u)_t-\Delta u\ni 0\quad \text{ in } D'(\Omega_T),\qquad \beta(s)=\begin{cases} s & \text{ for } s<0\cr [0,1] & \text{ for } s=0\cr 1+s & \text{ for } s>0. \end{cases} (2)_S \]
The exposé consists of five sections. Section 2 deals with the question of the regularity of the weak solutions of singular and degenerate quasilinear parabolic equations, proving their Hölder character. It starts with the precise definition of weak solution and the derivation of the building blocks of the theory: the local energy and logarithmic estimates. Further the authors briefly present the classical approach of De Giorgi to uniformly elliptic equations. De Giorgi’s class is introduced and it is shown that functions in De Giorgi’s class are Hölder continuous. In the same section the idea of intrinsic scaling is presented in full detail and, at least in the degenerate case, all the results leading to the Hölder continuity are proved. The theory is presented for the model case of the \(p\)-Laplace equation. The section is closed with remarks on the possible generalizations, namely to porous medium type equations. Section 3 addresses the boundedness of weak solutions. The theory discriminates between the degenerate and the singular case. If \(p>2,\) a local bound for the solution is implicit in the notion of weak solution. If \(1<p<2,\) local or global solutions need not be bounded in general.
Section 4 starts with a review of classical results concerning Harnack inequalities. The authors consider the degenerate case pointing out the differences with respect to the nondegenerate one. A proof of the Harnack inequality both in the degenerate and singular case is sketched. It is shown that for positive solutions of the singular \(p\)-Laplace equation an “elliptic” Harnack inequality holds. They also analyze the phenomenon of the extinction of the solution in finite time. Through a suitable use of the Raleigh quotient, sharp estimates on the extinction time are given and the asymptotic profile of the extinction is described. In the whole section the authors point out the major open questions on Harnack inequalities for singular and degenerate parabolic equations.
In Section 5 physical motivations are given concerning Stefan-like equations and, through the Kruzkov-Sukorjanski transformation the deep links between degenerate equations and Stefan-like equations are shown. The authors describe the approaches made by Aronson, Caffarelli, DiBenedetto, Sachs and Ziemer in the 1980s. They finally analyze the new pioneering approach of E. DiBenedetto and V. Vespri [Arch. Ration. Mech. Anal. 132, No. 3, 247–309 (1995; Zbl 0849.35060)] where the case of multiple singularities was totally solved in the case \(N=2.\) It is shown that this approach also works in the case \(N\geq 3\) but only under strong assumptions.
For the entire collection see [Zbl 1055.35006].