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Thierry Cazenave

Publications and source records attributed to Thierry Cazenave.

At least 19 recordsLinked to original sources
arXiv2020

Asymptotic behavior for a dissipative nonlinear Schrödinger equation

We consider the Schrödinger equation with nonlinear dissipation \begin{equation*} i \partial _t u +Δu=λ|u|^αu \end{equation*} in ${\mathbb R}^N $, $N\geq1$, where $λ\in {\mathbb C} $ with $\Imλ<0$. Assuming $\frac {2} {N+2}<α<\frac2N$, we give a precise description of the long-time behavior of the solutions (including decay rates in $L^2$ and $L^\infty $, and asymptotic profile), for a class of arbitrarily large initial data.

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arXiv2020

Sign-changing solutions of the nonlinear heat equation with persistent singularities

We study the existence of sign-changing solutions to the nonlinear heat equation $\partial _t u = Δu + |u|^αu$ on ${\mathbb R}^N $, $N\ge 3$, with $\frac {2} {N-2} < α<α_0$, where $α_0=\frac {4} {N-4+2\sqrt{ N-1 } }\in (\frac {2} {N-2}, \frac {4} {N-2})$, which are singular at $x=0$ on an interval of time. In particular, for certain $μ>0$ that can be arbitrarily large, we prove that for any $u_0 \in \mathrm{L} ^\infty _{\mathrm{loc}} ({\mathbb R}^N \setminus \{ 0 \}) $ which is bounded at infinity and equals $μ|x|^{- \frac {2} {α}}$ in a neighborhood of $0$, there exists a local (in time) solution $u$ of the nonlinear heat equation with initial value $u_0$, which is sign-changing, bounded at infinity and has the singularity $β|x|^{- \frac {2} {α}}$ at the origin in the sense that for $t>0$, $ |x|^{\frac {2} {α}} u(t,x) \to β$ as $ |x| \to 0$, where $β= \frac {2} {α} ( N -2 - \frac {2} {α} ) $. These solutions in general are neither stationary nor self-similar.

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arXiv2020

Sign-changing self-similar solutions of the nonlinear heat equation with positive initial value

We consider the nonlinear heat equation $u_t - Δu = |u|^αu$ on ${\mathbb R}^N$, where $α>0$ and $N\ge 1$. We prove that in the range $0 < α<\frac {4} {N-2}$, for every $μ>0$, there exist infinitely many sign-changing, self-similar solutions to the Cauchy problem with initial value $u_0 (x)= μ|x|^{-\frac {2} {α}}$. The construction is based on the analysis of the related inverted profile equation. In particular, we construct (sign-changing) self-similar solutions for positive initial values for which it is known that there does not exist any local, nonnegative solution.

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arXiv2020

Local smooth solutions of the nonlinear Klein-gordon equation

Given any $μ_1, μ_2\in {\mathbb C}$ and $α>0$, we prove the local existence of arbitrarily smooth solutions of the nonlinear Klein-Gordon equation $\partial_{ tt } u - Δu + μ_1 u = μ_2 |u|^αu$ on ${\mathbb R}^N$, $N\ge 1$, that do not vanish, i.e. $ |u (t,x) | >0 $ for all $x \in {\mathbb R}^N$ and all sufficiently small $t$. We write the equation in the form of a first-order system associated with a pseudo-differential operator, then use a method adapted from~[Commun. Contemp. Math. {\bf 19} (2017), no. 2, 1650038]. We also apply a similar (but simpler than in the case of the Klein-Gordon equation) argument to prove an analogous result for a class of nonlinear Dirac equations.

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arXiv2019

Solutions blowing up on any given compact set for the energy subcritical wave equation

We consider the focusing energy subcritical nonlinear wave equation $\partial_{tt} u - Δu= |u|^{p-1} u$ in ${\mathbb R}^N$, $N\ge 1$. Given any compact set $ E \subset {\mathbb R}^N $, we construct finite energy solutions which blow up at $t=0$ exactly on $ E$. The construction is based on an appropriate ansatz. The initial ansatz is simply $U_0(t,x) = κ(t + A(x) )^{ -\frac {2} {p-1} }$, where $A\ge 0$ vanishes exactly on $ E$, which is a solution of the ODE $h'' = h^p$. We refine this first ansatz inductively using only ODE techniques and taking advantage of the fact that (for suitably chosen $A$), space derivatives are negligible with respect to time derivatives. We complete the proof by an energy argument and a compactness method.

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arXiv2019

Solutions with prescribed local blow-up surface for the nonlinear wave equation

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 $\partial_{tt} u - Δu=|u|^{p-1} u$ on ${\mathbb R} \times {\mathbb R} ^N$, for any $1\leq N\leq 4$ 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 $H^2\times H^1$ solutions for the transformed problem.

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arXiv2019

Asymptotic behavior for a Schrödinger equation with nonlinear subcritical dissipation

We study the time-asymptotic behavior of solutions of the Schrödinger equation with nonlinear dissipation \begin{equation*} \partial _t u = i Δu + λ|u|^αu \end{equation*} in ${\mathbb R}^N $, $N\geq1$, where $λ\in {\mathbb C}$, $\Re λ<0$ and $0<α<\frac2N$. We give a precise description of the behavior of the solutions (including decay rates in $L^2$ and $L^\infty $, and asymptotic profile), for a class of arbitrarily large initial data, under the additional assumption that $α$ is sufficiently close to $\frac2N$.

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arXiv2019

Blowup on an arbitrary compact set for a Schödinger equation with nonlinear source term

We consider the nonlinear Schrödinger equation on ${\mathbb R}^N $, $N\ge 1$, \begin{equation*} \partial _t u = i Δu + λ| u |^αu \quad \mbox{on ${\mathbb R}^N $, $α>0$,} \end{equation*} with $λ\in {\mathbb C}$ and $\Re λ>0$, for $H^1$-subcritical nonlinearities, i.e. $α>0$ and $(N-2) α< 4$. 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 $U_0(t,x) = ( \Re λ)^{- \frac {1} {α}} (-αt + A(x) )^{ -\frac {1} {α} - i \frac {\Im λ} {α\Re λ} }$, where $A\ge 0$ vanishes exactly on $ K $, which is a solution of the ODE $u'= λ| u |^αu$. 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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arXiv2018

Finite-time blowup for a Schrödinger equation with nonlinear source term

We consider the nonlinear Schrödinger equation \[ u_t = i Δu + | u |^αu \quad \mbox{on ${\mathbb R}^N $, $α>0$,} \] for $H^1$-subcritical or critical nonlinearities: $(N-2) α\le 4$. Under the additional technical assumptions $α\geq 2$ (and thus $N\leq 4$), we construct $H^1$ solutions that blow up in finite time with explicit blow-up profiles and blow-up rates. In particular, blowup can occur at any given finite set of points of ${\mathbb R}^N$. The construction involves explicit functions $U$, solutions of the ordinary differential equation $U_t=|U|^αU$. In the simplest case, $U(t,x)=(|x|^k-αt)^{-\frac 1α}$ for $t<0$, $x\in {\mathbb R}^N$. For $k$ sufficiently large, $U$ satisfies $|ΔU|\ll U_t$ close to the blow-up point $(t,x)=(0,0)$, so that it is a suitable approximate solution of the problem. To construct an actual solution $u$ close to $U$, we use energy estimates and a compactness argument.

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arXiv2018

Perturbations of self-similar solutions

We consider the nonlinear heat equation $u_t = Δu + |u|^αu$ with $α>0$, either on ${\mathbb R}^N $, $N\ge 1$, or on a bounded domain with Dirichlet boundary conditions. We prove that in the Sobolev subcritical case $(N-2) α<4$, for every $μ\in {\mathbb R}$, if the initial value $u_0$ satisfies $u_0 (x) = μ|x-x_0|^{-\frac {2} {α}}$ in a neighborhood of some $x_0\in Ω$ and is bounded outside that neighborhood, then there exist infinitely many solutions of the heat equation with the initial condition $u(0)= u_0$. The proof uses a fixed-point argument to construct perturbations of self-similar solutions with initial value $μ|x-x_0|^{-\frac {2} {α}}$ on ${\mathbb R}^N $. Moreover, if $μ\ge μ_0$ for a certain $ μ_0( N, α)\ge 0$, and $u_0 I\ge 0$, then there is no nonnegative local solution of the heat equation with the initial condition $u(0)= u_0$, but there are infinitely many sign-changing solutions.

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arXiv2017

Modified scattering for the critical nonlinear Schrödinger equation

We consider the nonlinear Schrödinger equation $iu_t + Δu= λ|u|^{\frac {2} {N}} u $ in all dimensions $N\ge 1$, where $λ\in {\mathbb C}$ and $\Im λ\le 0$. We construct a class of initial values for which the corresponding solution is global and decays as $t\to \infty $, like $t^{- \frac {N} {2}}$ if $\Im λ=0$ and like $(t \log t)^{- \frac {N} {2}}$ if $\Im λ<0$. Moreover, we give an asymptotic expansion of those solutions as $t\to \infty $. We construct solutions that do not vanish, so as to avoid any issue related to the lack of regularity of the nonlinearity at $u=0$. To study the asymptotic behavior, we apply the pseudo-conformal transformation and estimate the solutions by allowing a certain growth of the Sobolev norms which depends on the order of regularity through a cascade of exponents.

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arXiv2016

Local existence, global existence, and scattering for the nonlinear Schrödinger equation

In this paper, we construct for every $α>0$ and $λ\in {\mathbb C}$ a space of initial values for which there exists a local solution of the nonlinear Schrödinger equation \begin{equation*} \begin{cases} iu_t + Δu + λ|u|^αu= 0 \\ u(0,x) = u_0 \end{cases} \end{equation*} on ${\mathbb R}^N $. Moreover, we construct for every $α>\frac {2} {N}$ a class of (arbitrarily large) initial values for which there exists a global solution that scatters as $t\to \infty $.

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arXiv2016

Finite-time blowup for some nonlinear complex Ginzburg-Landau equations

In this article, we review finite-time blowup criteria for the family of complex Ginzburg-Landau equations $u_t = e^{ iθ} [Δu + |u|^αu] + γu$ on ${\mathbb R}^N $, where $0 \le θ\le \frac {π} {2}$, $α>0$ and $γ\in {\mathbb R} $. We study in particular the effect of the parameters $θ$ and $γ$, and the dependence of the blowup time on these parameters.

math.AP↗
arXiv2015

A Fujita-type blowup result and low energy scattering for a nonlinear Schrö\-din\-ger equation

In this paper we consider the nonlinear Schrö\-din\-ger equation $i u_t +Δu +κ|u|^αu=0$. We prove that if $α<\frac {2} {N}$ and $\Im κ<0$, then every nontrivial $H^1$-solution blows up in finite or infinite time. In the case $α>\frac {2} {N}$ and $κ\in {\mathbb C}$, we improve the existing low energy scattering results in dimensions $N\ge 7$. More precisely, we prove that if $ \frac {8} {N + \sqrt{ N^2 +16N }} < α\le \frac {4} {N} $, then small data give rise to global, scattering solutions in $H^1$.

math.AP↗
arXiv2015

Non-regularity in Hölder and Sobolev spaces of solutions to the semilinear heat and Schrödinger equations

In this paper we study the Cauchy problem for the semilinear heat and Schrödinger equations, with the nonlinear term $ f ( u ) = λ|u|^αu$. We show that low regularity of $f$ (i.e., $α>0$ but small) limits the regularity of any possible solution for a certain class of smooth initial data. We employ two different methods, which yield two different types of results. On the one hand, we consider the semilinear equation as a perturbation of the ODE $w_t= f(w)$. This yields in particular an optimal regularity result for the semilinear heat equation in Hölder spaces. In addition, this approach yields ill-posedness results for NLS in certain $H^s$ spaces, which depend on the smallness of $α$ rather than the scaling properties of the equation. Our second method is to consider the semilinear equation as a perturbation of the linear equation via Duhamel's formula. This yields in particular that if $α$ is sufficiently small and $N$ sufficiently large, then the nonlinear heat equation is ill-posed in $H^s ({\mathbb R}^N ) $ for all $s\ge 0$.

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arXiv2013

Finite-time blowup for a complex Ginzburg-Landau equation with linear driving

In this paper, we consider the complex Ginzburg--Landau equation $u_t = e^{iθ} [Δu + |u|^αu] + γu$ on ${\mathbb R}^N $, where $α>0$, $γ\in \R$ and $-π/2<θ<π/2$. By convexity arguments we prove that, under certain conditions on $α,θ,γ$, a class of solutions with negative initial energy blows up in finite time.

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arXiv2013

Standing waves of the complex Ginzburg-Landau equation

We prove the existence of nontrivial standing wave solutions of the complex Ginzburg-Landau equation $ϕ_t = e^{iθ} Δϕ+ e^{iγ} |ϕ|^αϕ$ with periodic boundary conditions. Our result includes all values of $θ$ and $γ$ for which $\cos θ\cos γ>0$, but requires that $α>0$ be sufficiently small.

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