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Register a new userRefined blowup analysis and nonexistence of Type II blowups for an energy critical nonlinear heat equation
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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).
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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 several eigenvalues, with corresponding families of small standing waves, and we show that any small energy solution converges to the orbit of a time periodic solution plus a scattering term. The novel idea is to consider the refined profile, a quasi--periodic function in time which almost solves the NLS and encodes the discrete modes of a solution. The refined profile, obtained by elementary means, gives us directly an optimal coordinate system, avoiding the normal form arguments in cite{CM15APDE}, giving us also a better understanding of the Fermi Golden Rule.
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.
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 $u(0)in L^1_{rm{loc}}left({mathbb R}^Nsetminus{0}right)$, anti-symmetric with respect to $x_1,; x_2,; cdots,; x_m$ and $|u(0)|leq C(-1)^mpartial_{1}partial_{2}cdot cdot cdot partial_{m}(|x|^{-gamma})$ for $x_1>0,; cdots,; x_m>0,$ where $C>0$ is a constant, $min {1,; 2,; cdots,; N},$ $0<gamma<N$ and $0<alpha<2/(gamma+m).$ This gives a local existence result with highly singular initial values. As an application, for $a=1,$ we establish new blowup criteria for $0<alphaleq 2/(gamma+m)$, including the case $m=0.$ Moreover, if $(N-4)alpha<2,$ we prove the existence of initial values $u_0 = lambda f,$ for which the resulting solution blows up in finite time $T_{max}(lambda f),$ if $lambda>0$ is sufficiently small. We also construct blowing up solutions with initial data $lambda_n f$ such that $lambda_n^{[({1over alpha}-{gamma+mover 2})^{-1}]}T_{max}(lambda_n f)$ has different finite limits along different sequences $lambda_nto 0$. Our result extends the known small lambda blow up results for new values of $alpha$ and a new class of initial data.
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 $H^1$-subcritical nonlinearities, i.e. $alpha >0$ and $(N-2) alpha < 4$. Given a compact set $K subset {mathbb R}^N $, we construct $H^1$ solutions that are defined on $(-T,0)$ for some $T>0$, and blow up on $K $ at $t=0$. The construction is based on an appropriate ansatz. The initial ansatz is simply $U_0(t,x) = ( Re lambda )^{- frac {1} {alpha }} (-alpha t + A(x) )^{ -frac {1} {alpha } - i frac {Im lambda } {alpha Re lambda } }$, where $Age 0$ vanishes exactly on $ K $, which is a solution of the ODE $u= lambda | u |^alpha u$. We refine this ansatz inductively, using ODE techniques. We complete the proof by energy estimates and a compactness argument. This strategy is reminiscent of~[3, 4].
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