Standing waves of the complex Ginzburg-Landau equation

We prove the existence of nontrivial standing wave solutions of the complex Ginzburg-Landau equation $\phi_t = e^{i\theta} \Delta \phi + e^{i\gamma} |\phi |^\alpha \phi$ with periodic boundary conditions. Our result includes all values of $\theta$ and $\gamma$ for which $\cos \theta \cos \gamma >0$, but requires that $\alpha >0$ be sufficiently small

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

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

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

We consider the nonlinear heat equation $u_t - \Delta u = |u|^\alpha u$ on ${\mathbb R}^N$, where $\alpha >0$ and $N\ge 1$. We prove that in the range $0 0$, there exist infinitely many sign-changing, self-similar solutions to the Cauchy problem with initial value $u_0 (x)= \mu |x|^{-\frac {2} {\alpha }}$. 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

Finite-time blowup for a complex Ginzburg-Landau equation

We prove that negative energy solutions of the complex Ginzburg-Landau equation $e^{-i\theta} u_t = \Delta u+ |u|^{\alpha} u$ blow up in finite time, where \alpha >0 and \pi /2<\theta <\pi /2. For a fixed initial value $u(0)$, we obtain estimates of the blow-up time $T_{max}^\theta$ as $\theta \to \pm \pi /2$. It turns out that $T_{max}^\theta$ stays bounded (respectively, goes to infinity) as $\theta \to \pm \pi /2$ in the case where the solution of the limiting nonlinear Schr\"odinger equation blows up in finite time (respectively, is global).Comment: 22 page

Sharp conditions for blowup of solutions of a chemotactical model for two species in R2

AbstractWe consider a model system of Keller–Segel type for the evolution of two species in the whole space R2 which are driven by chemotaxis and diffusion. It is well known that this problem admits global and blowup solutions. We show that there exists a sharp condition which allows to distinguish global and blowup solutions in the radially symmetric case. More precisely, let m∞ and n∞ be the total masses of the species. Then there exists a critical curve γ in the m∞−n∞ plane such that the solution blows up if and only if (m∞,n∞) is above γ. This gives an answer to a question raised by Conca et al. (2011) in [8]. We also study the asymptotic behaviour of global solutions in the subcritical case, showing that they are asymptotically self-similar

Implicit finite difference schemes for a linear model of well-reservoir coupling

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Universal solutions of the heat equation on RN\mathbb R^N

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

International audienceIn this paper we study the Cauchy problem for the semilinear heat and Schrödinger equations, with the nonlinear term f (u) = λ|u| α u. 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 wt = 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 (R N) for all s ≥ 0