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Register a new userBubble towers in the ancient solution of energy-critical heat equation
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We construct a radially smooth positive ancient solution for energy critical semi-linear heat equation in $mathbb{R}^n$, $ngeq 7$. It blows up at the origin with the profile of multiple Talenti bubbles in the backward time infinity.
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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).
We consider the generic divergence form second order parabolic equation with coefficients that are regular in the spatial variables and just measurable in time. We show that the spatial derivatives of its fundamental solution admit upper bounds that agree with the Aronson type estimate and only depend on the ellipticity constants of the equation and the L $infty$ norm of the spatial derivatives of its coefficients. We also study the corresponding stochastic partial differential equations and prove that under natural assumptions on the noise the equation admits a mild solution, given by anticipating stochastic integration.
We give here an explicit formula for the following critical case of the growth-fragmentation equation $$frac{partial}{partial t} u(t, x) + frac{partial}{partial x} (gxu(t, x)) + bu(t, x) = balpha^2 u(t, alpha x), qquad u(0, x) = u_0 (x),$$ for some constants $g > 0$, $b > 0$ and $alpha > 1$ - the case $alpha = 2$ being the emblematic binary fission case. We discuss the links between this formula and the asymptotic ones previously obtained in (Doumic, Escobedo, Kin. Rel. Mod., 2016), and use them to clarify how periodicity may appear asymptotically.
We consider the energy-critical non-linear focusing wave equation in dimension N=3,4,5. An explicit stationnary solution, $W$, of this equation is known. The energy E(W,0) has been shown by C. Kenig and F. Merle to be a threshold for the dynamical behavior of solutions of the equation. In the present article we study the dynamics at the critical level E(u_0,u_1)=E(W,0) and classify the corresponding solutions. We show in particular the existence of two special solutions, connecting different behaviors for negative and positive times. Our results are analoguous to our previous work on radial Schrodinger equation, but without any radial assumption on the data. We also refine the understanding of the dynamical behavior of the special solutions.
We consider the semilinear heat equation, to which we add a nonlinear gradient term, with a critical power. We construct a solution which blows up in finite time. We also give a sharp description of its blow-up profile. The proof relies on the reduction of the problem to a finite dimensional one, and uses the index theory to conclude. Thanks to the interpretation of the parameters of the finite-dimensional problem in terms of the blow-up time and point, we also show the stability of the constructed solution with respect to initial data. This note presents the results and the main arguments. For the details, we refer to our paper cite{TZ15}.
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