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Register a new userNew type II Finite time blow-up for the energy supercritical heat equation
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We consider the energy supercritical heat equation with the $(n-3)$-th Sobolev exponent begin{equation*} begin{cases} u_t=Delta u+u^{3},~&mbox{ in } Omegatimes (0,T), u(x,t)=u|_{partialOmega},~&mbox{ on } partialOmegatimes (0,T), u(x,0)=u_0(x),~&mbox{ in } Omega, end{cases} end{equation*} where $5leq nleq 7$, $Omega=R^n$ or $Omega subset R^n$ is a smooth, bounded domain enjoying special symmetries. We construct type II finite time blow-up solution $u(x,t)$ with the singularity taking place along an $(n-4)$-dimensional {em shrinking sphere} in $Omega$. More precisely, at leading order, the solution $u(x,t)$ is of the sharply scaled form $$u(x,t)approx la^{-1}(t)frac{2sqrt{2}}{1+left|frac{(r,z)-(xi_r(t),xi_z(t))}{la(t)}right|^2}$$ where $r=sqrt{x_1^2+cdots+x_{n-3}^2}$, $z=(x_{n-2},x_{n-1},x_n)$ with $x=(x_1,cdots,x_n)inOmega$. Moreover, the singularity location $$(xi_r(t),xi_z(t))sim (sqrt{2(n-4)(T-t)},z_0)~mbox{ as }~t earrow T,$$ for some fixed $z_0$, and the blow-up rate $$la(t)sim frac{T-t}{|log(T-t)|^2}~mbox{ as }~t earrow T.$$ This is a completely new phenomenon in the parabolic setting.
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Consider the energy critical focusing wave equation on the Euclidian space. A blow-up type II solution of this equation is a solution which has finite time of existence but stays bounded in the energy space. The aim of this work is to exhibit universal properties of such solutions. Let W be the unique radial positive stationary solution of the equation. Our main result is that in dimension 3, under an appropriate smallness assumption, any type II blow-up radial solution is essentially the sum of a rescaled W concentrating at the origin and a small remainder which is continuous with respect to the time variable in the energy space. This is coherent with the solutions constructed by Krieger, Schlag and Tataru. One ingredient of our proof is that the unique radial solution which is compact up to scaling is equal to W up to symmetries.
Following our previous paper in the radial case, we consider blow-up type II solutions to the energy-critical focusing wave equation. Let W be the unique radial positive stationary solution of the equation. Up to the symmetries of the equation, under an appropriate smallness assumption, any type II blow-up solution is asymptotically a regular solution plus a rescaled Lorentz transform of W concentrating at the origin.
Let $OmegasubsetR^n$ be a smooth bounded domain and let $a_1,a_2,dots,a_{i_0}inOmega$, $widehat{Omega}=Omegasetminus{a_1,a_2,dots,a_{i_0}}$ and $widehat{R^n}=R^nsetminus{a_1,a_2,dots,a_{i_0}}$. We prove the existence of solution $u$ of the fast diffusion equation $u_t=Delta u^m$, $u>0$, in $widehat{Omega}times (0,infty)$ ($widehat{R^n}times (0,infty)$ respectively) which satisfies $u(x,t)toinfty$ as $xto a_i$ for any $t>0$ and $i=1,cdots,i_0$, when $0<m<frac{n-2}{n}$, $ngeq 3$, and the initial value satisfies $0le u_0in L^p_{loc}(2{Omega}setminus{a_1,cdots,a_{i_0}})$ ($u_0in L^p_{loc}(widehat{R^n})$ respectively) for some constant $p>frac{n(1-m)}{2}$ and $u_0(x)ge lambda_i|x-a_i|^{-gamma_i}$ for $xapprox a_i$ and some constants $gamma_i>frac{2}{1-m},lambda_i>0$, for all $i=1,2,dots,i_0$. We also find the blow-up rate of such solutions near the blow-up points $a_1,a_2,dots,a_{i_0}$, and obtain the asymptotic large time behaviour of such singular solutions. More precisely we prove that if $u_0gemu_0$ on $widehat{Omega}$ ($widehat{R^n}$, respectively) for some constant $mu_0>0$ and $gamma_1>frac{n-2}{m}$, then the singular solution $u$ converges locally uniformly on every compact subset of $widehat{Omega}$ (or $widehat{R^n}$ respectively) to infinity as $ttoinfty$. If $u_0gemu_0$ on $widehat{Omega}$ ($widehat{R^n}$, respectively) for some constant $mu_0>0$ and satisfies $lambda_i|x-a_i|^{-gamma_i}le u_0(x)le lambda_i|x-a_i|^{-gamma_i}$ for $xapprox a_i$ and some constants $frac{2}{1-m}<gamma_ilegamma_i<frac{n-2}{m}$, $lambda_i>0$, $lambda_i>0$, $i=1,2,dots,i_0$, we prove that $u$ converges in $C^2(K)$ for any compact subset $K$ of $2{Omega}setminus{a_1,a_2,dots,a_{i_0}}$ (or $widehat{R^n}$ respectively) to a harmonic function as $ttoinfty$.
Let $nge 3$ and $0<m<frac{n-2}{n}$. We will extend the results of J.L. Vazquez and M. Winkler and prove the uniqueness of finite points blow-up solutions of the fast diffusion equation $u_t=Delta u^m$ in both bounded domains and $mathbb{R}^ntimes (0,infty)$. We will also construct initial data such that the corresponding solution of the fast diffusion equation in bounded domain oscillate between infinity and some positive constant as $ttoinfty$.
We give a sufficient condition for non-existence of global nonnegative mild solutions of the Cauchy problem for the semilinear heat equation $u = Lu + f(u)$ in $L^p(X,m)$ for $p in [1,infty)$, where $(X,m)$ is a $sigma$-finite measure space, $L$ is the infinitesimal generator of a sub-Markovian strongly continuous semigroup of bounded linear operators in $L^p(X,m)$, and $f$ is a strictly increasing, convex, continuous function on $[0,infty)$ with $f(0) = 0$ and $int_1^infty 1/f < infty$. Since we make no further assumptions on the behaviour of the diffusion, our main result can be seen as being about the competition between the diffusion represented by $L$ and the reaction represented by $f$ in a general setting. We apply our result to Laplacians on manifolds, graphs, and, more generally, metric measure spaces with a heat kernel. In the process, we recover and extend some older as well as recent results in a unified framework.
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