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We consider the energy critical semilinear heat equation $$ left{begin{aligned} &partial_t u-Delta u =|u|^{frac{4}{n-2}}u &mbox{in } {mathbb R}^ntimes(0,T), &u(x,0)=u_0(x), end{aligned}right. $$ where $ ngeq 3$, $u_0in L^infty({mathbb R}^n)$, and $Tin {mathbb R}^+$ is the first blow up time. We prove that if $ n geq 7$ and $ u_0 geq 0$, then any blowup must be of Type I, i.e., [|u(cdot, t)|_{L^infty({mathbb R}^n)}leq C(T-t)^{-frac{1}{p-1}}.] A similar result holds for bounded convex domains. The proof relies on a reverse inner-outer gluing mechanism and delicate analysis of bubbling behavior (bubbling tower/cluster).
In this paper we give a new and simplified proof of the theorem on selection of standing waves for small energy solutions of the nonlinear Schrodinger equations (NLS) that we gave in cite{CM15APDE}. We consider a NLS with a Schrodinger operator with
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 this paper we prove local existence of solutions to the nonlinear heat equation $u_t = Delta u +a |u|^alpha u, ; tin(0,T),; x=(x_1,,cdots,, x_N)in {mathbb R}^N,; a = pm 1,; alpha>0;$ with initial value $u(0)in L^1_{rm{loc}}left({mathbb R}^Nsetminu
We establish Strichartz estimates for the radial energy-critical wave equation in 5 dimensions in similarity coordinates. Using these, we prove the nonlinear asymptotic stability of the ODE blowup in the energy space.
We consider the nonlinear Schrodinger equation on ${mathbb R}^N $, $Nge 1$, begin{equation*} partial _t u = i Delta u + lambda | u |^alpha u quad mbox{on ${mathbb R}^N $, $alpha>0$,} end{equation*} with $lambda in {mathbb C}$ and $Re lambda >0$, for