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Register a new userUniversality of blow-up profile for small radial type II blow-up solutions of energy-critical wave equation
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Authors
Thomas Duyckaerts
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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.
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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.
We describe the asymptotic behavior of positive solutions $u_epsilon$ of the equation $-Delta u + au = 3,u^{5-epsilon}$ in $Omegasubsetmathbb{R}^3$ with a homogeneous Dirichlet boundary condition. The function $a$ is assumed to be critical in the sense of Hebey and Vaugon and the functions $u_epsilon$ are assumed to be an optimizing sequence for the Sobolev inequality. Under a natural nondegeneracy assumption we derive the exact rate of the blow-up and the location of the concentration point, thereby proving a conjecture of Brezis and Peletier (1989). Similar results are also obtained for solutions of the equation $-Delta u + (a+epsilon V) u = 3,u^5$ in $Omega$.
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.
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.
We prove that the critical Wave Maps equation with target $S^2$ and origin $mathbb{R}^{2+1}$ admits energy class blow up solutions of the form $$u(t,r)=Q(lambda(t)r)+epsilon(t,r)$$where $Q: mathbb{R}^2 to S^2$ is the ground state harmonic map and $lambda(t) = t^{-1- u}$ for any $ u > 0$. This extends the work [13], where such solutions were constructed under the assumption $ u > 1/2$. In light of a result of Struwe [22], our result is optimal for polynomial blow up rates.
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