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Register a new userBubbling and extinction for some fast diffusion equations in bounded domains
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We study a Sobolev critical fast diffusion equation in bounded domains with the Brezis-Nirenberg effect. We obtain extinction profiles of its positive solutions, and show that the convergence rates of the relative error in regular norms are at least polynomial. Exponential decay rates are proved for generic domains. Our proof makes use of its regularity estimates, a curvature type evolution equation, as well as blow up analysis. Results for Sobolev subcritical fast diffusion equations are also obtained.
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We prove optimal boundary regularity for bounded positive weak solutions of fast diffusion equations in smooth bounded domains. This solves a problem raised by Berryman and Holland in 1980 for these equations in the subcritical and critical regimes. Our proof of the a priori estimates uses a geometric type structure of the fast diffusion equations, where an important ingredient is an evolution equation for a curvature-like quantity.
We study extinction profiles of solutions to fast diffusion equations with some initial data in the Marcinkiewicz space. The extinction profiles will be the singular solutions of their stationary equations.
In the present paper, we investigate the regularity and symmetry properties of weak solutions to semilinear elliptic equations which are locally stable.
The dissipative wave equation with a critical quintic nonlinearity in smooth bounded three dimensional domain is considered. Based on the recent extension of the Strichartz estimates to the case of bounded domains, the existence of a compact global attractor for the solution semigroup of this equation is established. Moreover, the smoothness of the obtained attractor is also shown.
Consider the integral equation begin{equation*} f^{q-1}(x)=int_Omegafrac{f(y)}{|x-y|^{n-alpha}}dy, f(x)>0,quad xin overline Omega, end{equation*} where $Omegasubset mathbb{R}^n$ is a smooth bounded domain. For $1<alpha<n$, the existence of energy maximizing positive solution in subcritical case $2<q<frac{2n}{n+alpha}$, and nonexistence of energy maximizing positive solution in critical case $q=frac{2n}{n+alpha}$ are proved in cite{DZ2017}. For $alpha>n$, the existence of energy minimizing positive solution in subcritical case $0<q<frac{2n}{n+alpha}$, and nonexistence of energy minimizing positive solution in critical case $q=frac{2n}{n+alpha}$ are also proved in cite{DGZ2017}. Based on these, in this paper, the blowup behaviour of energy maximizing positive solution as $qto (frac{2n}{n+alpha})^+ $ (in the case of $1<alpha<n$), and the blowup behaviour of energy minimizing positive solution as $qto (frac{2n}{n+alpha})^-$ (in the case of $alpha>n$) are analyzed. We see that for $1<alpha<n$ the blowup behaviour obtained is quite similar to that of the elliptic equation involving subcritical Sobolev exponent. But for $alpha>n$, different phenomena appears.
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