The regularity of generalized solutions for the n-dimensional quasi-linear parabolic diffraction problems

In this paper, we study the regularity of generalized solutions u(x,t)$u(x,t)$ for the n  -dimensional quasi-linear parabolic diffraction problem. By using various estimates and Steklov average methods, we prove that (1): for almost all t$t$ the first derivatives _ux(x,t)$u_x(x,t)$ are Hölder continuous with respect to x$x$ up to the inner boundary, on which the coefficients of the equation are allowed to be discontinuous; and (2): the first derivative _ut(x,t)$u_t(x,t)$ is Hölder continuous with respect to (x,t)$(x,t)$ across the inner boundary.     

Critical exponents for non-simultaneous blow-up in a localized parabolic system

The paper deals with the radially symmetric solutions of _ut=Δu+u^m(x,t)vⁿ(0,t)$u_t=\Delta u+u^m(x,t)v^n(0,t)$, _vt=Δv+u^p(0,t)v^q(x,t)$v_t=\Delta v+u^p(0,t)v^q(x,t)$, subject to null Dirichlet boundary conditions. For the blow-up classical solutions, we propose the critical exponents for non-simultaneous blow-up by determining the complete and optimal classification for all the non-negative exponents: (i) There exist initial data such that u$u$ (v$v$) blows up alone if and only if m>p+1$m>p+1$ (q>n+1$q>n+1$), which means that any blow-up is simultaneous if and only if m≤p+1$m\le p+1$, q≤n+1$q\le n+1$. (ii) Any blow-up is u$u$ (v$v$) blowing up with v$v$ (u$u$) remaining bounded if and only if m>p+1$m>p+1$, q≤n+1$q\le n+1$ (m≤p+1$m\le p+1$, q>n+1$q>n+1$). (iii) Both non-simultaneous and simultaneous blow-up may occur if and only if m>p+1$m>p+1$, q>n+1$q>n+1$. Moreover, we consider the blow-up rate and set estimates which were not obtained in the previously known work for the same model.