Blowup solutions of Grushin's operator

In this note, we consider the blowup phenomenon of Grushin's operator. By using the knowledge of probability, we first get expression of heat kernel of Grushin's operator. Then by using the properties of heat kernel and suitable auxiliary function, we get that the solutions will blow up in finite time.Comment:

Blow-up properties for a degenerate parabolic system with nonlinear localized sources

AbstractThis paper deals with blow-up properties for a degenerate parabolic system with nonlinear localized sources subject to the homogeneous Dirichlet boundary conditions. The main aim of this paper is to study the blow-up rate estimate and the uniform blow-up profile of the blow-up solution. Our conclusions extend the results of [L.L. Du, Blow-up for a degenerate reaction–diffusion system with nonlinear localized sources, J. Math. Anal. Appl. 324 (2006) 304–320]. At the end, the blow-up set and blow up rate with respect to the radial variable is considered when the domain Ω is a ball

Global solutions and blow-up problems for a nonlinear degenerate parabolic system coupled via nonlocal sources

AbstractThis paper concerns with a nonlinear degenerate parabolic system coupled via nonlocal sources, subjecting to homogeneous Dirichlet boundary condition. The main aim of this paper is to study conditions on the global existence and/or blow-up in finite time of solutions, and give the estimates of blow-up rates of blow-up solutions

Nonlinear stability of source defects in oscillatory media

In this paper, we prove the nonlinear stability under localized perturbations of spectrally stable time-periodic source defects of reaction-diffusion systems. Consisting of a core that emits periodic wave trains to each side, source defects are important as organizing centers of more complicated flows. Our analysis uses spatial dynamics combined with an instantaneous phase-tracking technique to obtain detailed pointwise estimates describing perturbations to lowest order as a phase-shift radiating outward at a linear rate plus a pair of localized approximately Gaussian excitations along the phase-shift boundaries; we show that in the wake of these outgoing waves the perturbed solution converges time-exponentially to a space-time translate of the original source pattern.https://arxiv.org/abs/1802.07676First author draf

Blow-up of Solution for Initial Boundary Value Problem of Reaction Diffusion Equations

In this paper, the blow-up of solution for the initial boundary value problem of a class of reaction diffusion equations with multiple nonlinearities is studied. We prove, under suitable conditions on memory and nonlinearities term and for negative or positive initial energy, a global nonexistence theorem