Five types of blow-up in a semilinear fourth-order reaction-diffusion equation: an analytic-numerical approach

Five types of blow-up patterns that can occur for the 4th-order semilinear parabolic equation of reaction-diffusion type u_t= -\Delta^2 u + |u|^{p-1} u \quad {in} \quad \ren \times (0,T), p>1, \quad \lim_{t \to T^-}\sup_{x \in \ren} |u(x,t)|= +\iy, are discussed. For the semilinear heat equation $u_t= \Delta u+ u^p$, various blow-up patterns were under scrutiny since 1980s, while the case of higher-order diffusion was studied much less, regardless a wide range of its application.Comment: 41 pages, 27 figure

Phase Space Models for Stochastic Nonlinear Parabolic Waves: Wave Spread and Singularity

We derive several kinetic equations to model the large scale, low Fresnel number behavior of the nonlinear Schrodinger (NLS) equation with a rapidly fluctuating random potential. There are three types of kinetic equations the longitudinal, the transverse and the longitudinal with friction. For these nonlinear kinetic equations we address two problems: the rate of dispersion and the singularity formation. For the problem of dispersion, we show that the kinetic equations of the longitudinal type produce the cubic-in-time law, that the transverse type produce the quadratic-in-time law and that the one with friction produces the linear-in-time law for the variance prior to any singularity. For the problem of singularity, we show that the singularity and blow-up conditions in the transverse case remain the same as those for the homogeneous NLS equation with critical or supercritical self-focusing nonlinearity, but they have changed in the longitudinal case and in the frictional case due to the evolution of the Hamiltonian

Nonlinear Convection in Reaction-diffusion Equations under dynamical boundary conditions

We investigate blow-up phenomena for positive solutions of nonlinear reaction-diffusion equations including a nonlinear convection term $\partial_t u = \Delta u - g(u) \cdot \nabla u + f(u)$ in a bounded domain of $\mathbb{R}^N$ under the dissipative dynamical boundary conditions $\sigma \partial_t u + \partial_\nu u =0$. Some conditions on $g$ and $f$ are discussed to state if the positive solutions blow up in finite time or not. Moreover, for certain classes of nonlinearities, an upper-bound for the blow-up time can be derived and the blow-up rate can be determinated.Comment: 20 page

Boundary fluxes for non-local diffusion

We study a nonlocal diffusion operator in a bounded smooth domain prescribing the flux through the boundary. This problem may be seen as a generalization of the usual Neumann problem for the heat equation. First, we prove existence, uniqueness and a comparison principle. Next, we study the behavior of solutions for some prescribed boundary data including blowing up ones. Finally, we look at a nonlinear flux boundary condition

Nonlinear Schr\"odinger Equation with a White-Noise Potential: Phase-space Approach to Spread and Singularity

We propose a phase-space formulation for the nonlinear Schr\"odinger equation with a white-noise potential in order to shed light on two issues: the rate of spread and the singularity formation in the average sense. Our main tools are the energy law and the variance identity. The method is completely elementary. For the problem of wave spread, we show that the ensemble-averaged dispersion in the critical or defocusing case follows the cubic-in-time law while in the supercritical and subcritical focusing cases the cubic law becomes an upper and lower bounds respectively. We have also found that in the critical and supercritical focusing cases the presence of a white-noise random potential results in different conditions for singularity-with-positive-probability from the homogeneous case but does not prevent singularity formation. We show that in the supercritical focusing case the ensemble-averaged self-interaction energy and the momentum variance can exceed any fixed level in a finite time with positive probability