Related papers: Blow-Up for Nonlinear Wave Equations describing Bo…
Consider the nonlinear heat equation $v_t -\Delta v= |v|^{p-1} v$ in a bounded smooth domain $\Omega\subset \R^n$ with $n>2$ and Dirichlet boundary condition. Given $u_{p}$ a sign-changing stationary solution fulfilling suitable…
Fourth-order semilinear parabolic equations of the Cahn--Hilliard-type (01) u_t + \D^2 u = \g u \pm \D (|u|^{p-1}u) in \Omega \times \re_+, are considered in a smooth bounded domain $\O \subset \ren$ with Navier-type boundary conditions on…
We study the focusing inhomogeneous nonlinear Schr\"odinger equation $$ i\partial_t u + \Delta u = -|x|^b |u|^{p-1}u ,\quad (t,x)\in (0,\infty)\times\mathbb{R}^N, $$ with $b>0$ and $p>1$. Due to the spatial growth of the nonlinearity,…
We consider blow-up solutions of a semilinear wave equation with a loglog perturbation of the power nonlinearity in the subconformal case, and show that the blow-up rate is given by the solution of the associated ODE which has the same…
This paper is concerned with the long-time dynamics of the nonlinear wave equation in one-space dimension, $$ u_{tt} - \delta^2 u_{xx} +V'(u) =0 \qquad x\in [0,1] $$ where $\delta>0$ is a parameter and $V(u)$ is a potential bounded from…
We undertake a comprehensive study of the nonlinear Schr\"odinger equation $$ i u_t +\Delta u = \lambda_1|u|^{p_1} u+ \lambda_2 |u|^{p_2} u, $$ where $u(t,x)$ is a complex-valued function in spacetime $\R_t\times\R^n_x$, $\lambda_1$ and…
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 $0<\varepsilon\ll 1$ . We will prove the existence and stability of a blow-up…
We consider the initial boundary value problem in exterior domain for strongly damped wave equations with power type nonlinearity |u|^p. We will establish blow-up results under some conditions on the initial data and the exponent p.
In this paper we will continue the analysis of two dimensional Schr\"odinger equation with a fixed, pointwise, nonlinearity started in [2, 13]. In this model, the occurrence of a blow-up phenomenon has two peculiar features: the energy…
We consider the focusing NLS with an angular momentum and a harmonic potential, which models Bose-Einstein condensate under a rotating magnetic trap. We give a sharp condition on the global existence and blowup in the mass-critical case. We…
Consider the stochastic partial differential equation u_t=u_{xx}+u^gamma dot{W}, where x in [0,J], dot{W}=dot{W}(t,x) is 2-parameter white noise, and we assume that the initial function u(0,x) is nonnegative and not identically 0. We impose…
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…
We investigate the nonisospectral effects of a semi-discrete nonlinear Schr\"{o}dinger equation, which is a direct integrable discretisation of its continuous counterpart. Bilinear form and double casoratian solution of the equation are…
Existence of finite-time blow ups in the classical one-dimensional nonlinear Schr\"odinger equation (NLS) (1) i \partial_t u + u_{x x} + |u|^{2r} u = 0, u(x,0) = u_0(x) has been one of the central problems in the studies of the singularity…
We study both analytically and numerically the nonlinear stage of the instability of one-dimensional solitons in a small vicinity of the transition point from supercritical to subcritical bifurcations in the framework of the generalized…
We study in this article the blow-up of solutions to a coupled semilinear wave equations which are characterized by linear damping terms in the \textit{scale-invariant regime}, time-derivative nonlinearities, mass terms and Tricomi terms.…
We prove finite-time Type-I blowup for the three-dimensional incompressible Euler equations in the axisymmetric no-swirl class, with initial velocity in $C^{1,\alpha}(\mathbb{R}^3)\cap L^2(\mathbb{R}^3)$, odd symmetry in $z$, and…
In this article, we study the blowup phenomena of compressible Euler equations with non-vacuum initial data. Our new results, which cover a general class of testing functions, present new initial value blowup conditions. The corresponding…
In this paper, we study the blow-up of solutions for semilinear wave equations with scale-invariant dissipation and mass in the case in which the model is somehow 'wave-like'. A Strauss type critical exponent is determined as the upper…
The problem of blow up of solutions to the initial boundary value problem for non-autonomous semilinear wave equation with damping and accelerating terms under the Robin boundary condition is studied. Sufficient conditions of blow up in a…