相关论文: Construction of blowup solutions for the Complex G…
We construct a solution for the Complex Ginzburg-Landau equation in some critical case, which blows up in finite time $T$ only at one blow-up point. We also give a sharp description of its profile. The proof relies on the reduction of the…
In this paper, we consider the complex Ginzburg--Landau equation $u_t = e^{i\theta} [\Delta u + |u|^\alpha u] + \gamma u$ on ${\mathbb R}^N $, where $\alpha >0$, $\gamma \in \R$ and $-\pi /2<\theta <\pi /2$. By convexity arguments we prove…
In this article, we review finite-time blowup criteria for the family of complex Ginzburg-Landau equations $u_t = e^{ i\theta } [\Delta u + |u|^\alpha u] + \gamma u$ on ${\mathbb R}^N $, where $0 \le \theta \le \frac {\pi } {2}$, $\alpha…
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…
We construct finite time blow-up solutions to the Landau-Lifshitz-Gilbert equation (LLG) from ${\mathbb R}^2$ into $S^2$ \begin{equation*} \begin{cases} u_t= a(\Delta u+|\nabla u|^2u) -b u\wedge \Delta u &\ \mbox{ in }\ {\mathbb…
We construct a solution to a complex nonlinear heat equation which blows up in finite time $T$ only at one blow-up point. We also give a sharp description of its blow-up profile. The proof relies on the reduction of the problem to a finite…
In the last twenty years, there have been significant advances in the study of the blow-up phenomenon for the critical generalized Korteweg-de Vries equation, including the determination of sufficient conditions for blowup, the stability of…
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…
We consider the following exponential reaction-diffusion equation involving a nonlinear gradient term: $$\partial_t U = \Delta U + \alpha|\nabla U|^2 + e^U,\quad (x, t)\in\mathbb{R}^N\times[0,T), \quad \alpha > -1.$$ We construct for this…
In this paper, we consider the complex Ginzburg-Landau equation $$ \partial_t u = (1 + i \beta) \Delta u + (1 + i \delta) |u|^{p-1}u - \alpha u, \quad \text{where } \beta, \delta, \alpha \in \mathbb{R}. $$ The study focuses on investigating…
We consider the nonlinear Schr\"odinger equation \[ u_t = i \Delta u + | u |^\alpha u \quad \mbox{on ${\mathbb R}^N $, $\alpha>0$,} \] for $H^1$-subcritical or critical nonlinearities: $(N-2) \alpha \le 4$. Under the additional technical…
We construct a two-parameter continuum of type II blow up solutions for the energy-critical focusing NLS in dimension $ d = 3$. The solutions collapse to a single energy bubble in finite time, precisely they have the form $ u(t,x) = e^{i…
In this paper, we consider the finite time blow-up results for a parabolic equation coupled with superlinear source term and local linear boundary dissipation. Using a concavity argument, we derive the sufficient conditions for the…
We consider the energy critical four dimensional semi linear heat equation \partial tu-\Deltau-u3 = 0. We show the existence of type II finite time blow up solutions and give a sharp description of the corresponding singularity formation.…
In this paper we consider the slightly $L^2$-supercritical gKdV equations $\partial_t u+(u_{xx}+u|u|^{p-1})_x=0$, with the nonlinearity $5<p<5+\varepsilon$ and $0<\varepsilon\ll 1$ . In the previous work of the author we know that there…
We construct a solution for a class of strongly perturbed semilinear heat equations which blows up in finite time with a prescribed blow-up profile. The construction relies on the reduction of the problem to a finite dimensional one and the…
In this paper, we are concerned with complex Ginzburg-Landau (CGL) equations. There are several results on the global existence and smoothing effects of solutions to the initial boundary value problem for (CGL) in bounded or unbounded…
We address the critical norm blow-up problem for the nonlinear heat equation $u_t-\Delta u=|u|^{p-1}u$ in $\mathbf{R}^n\times(0,T)$. In the supercritical range $p>(n+2)/(n-2)$, we prove that if the maximal existence time $T$ is finite, then…
The study of blow-up solution of time-fractional heat equations is of significant and wide-ranging interest for its multitude of applications. These types of equations are used to model several real problems in science and engineering. This…
In this paper we develop two different types of criteria for the finite time blow-up solutions to the combined nonlinear Schr\"odinger equation in 1D. The first one is a negative energy criterion developed for triple combined nonlinearity…