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Register a new userExistence of a Stable Blow-up profile for the nonlinear heat equation with a critical power nonlinear gradient term
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We consider the nonlinear heat equation with a nonlinear gradient term: $partial_t u =Delta u+mu| abla u|^q+|u|^{p-1}u,; mu>0,; q=2p/(p+1),; p>3,; tin (0,T),; xin R^N.$ We construct a solution which blows up in finite time $T>0.$ We also give a sharp description of its blow-up profile and show that it is stable with respect to perturbations in initial data. The proof relies on the reduction of the problem to a finite dimensional one, and uses the index theory to conclude. The blow-up profile does not scale as $(T-t)^{1/2}|log(T-t)|^{1/2},$ like in the standard nonlinear heat equation, i.e. $mu=0,$ but as $(T-t)^{1/2}|log(T-t)|^{beta}$ with $beta=(p+1)/[2(p-1)]>1/2.$ We also show that $u$ and $ abla u$ blow up simultaneously and at a single point, and give the final profile. In particular, the final profile is more singular than the case of the standard nonlinear heat equation.
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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}.
We prove that any sufficiently differentiable space-like hypersurface of ${mathbb R}^{1+N} $ coincides locally around any of its points with the blow-up surface of a finite-energy solution of the focusing nonlinear wave equation $partial_{tt} u - Delta u=|u|^{p-1} u$ on ${mathbb R} times {mathbb R} ^N$, for any $1leq Nleq 4$ and $1 < p le frac {N+2} {N-2}$. We follow the strategy developed in our previous work [arXiv 1812.03949] on the construction of solutions of the nonlinear wave equation blowing up at any prescribed compact set. Here to prove blowup on a local space-like hypersurface, we first apply a change of variable to reduce the problem to blowup on a small ball at $t=0$ for a transformed equation. The construction of an appropriate approximate solution is then combined with an energy method for the existence of a solution of the transformed problem that blows up at $t=0$. To obtain a finite-energy solution of the original problem from trace arguments, we need to work with $H^2times H^1$ solutions for the transformed problem.
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.
The blowup is studied for the nonlinear Schr{o}dinger equation $iu_{t}+Delta u+ |u|^{p-1}u=0$ with $p$ is odd and $pge 1+frac 4{N-2}$ (the energy-critical or energy-supercritical case). It is shown that the solution with negative energy $E(u_0)<0$ blows up in finite or infinite time. A new proof is also presented for the previous result in cite{HoRo2}, in which a similar result but more general in a case of energy-subcritical was shown.
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.
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