Explosive solutions of parabolic stochastic partial differential equations with lévy noise

Kexue Li,Jigen Peng,Junxiong Jia

doi:10.3934/dcds.2017221

Abstract

In this paper, we study the explosive solutions to a class of parbolic stochastic semilinear differential equations driven by a Levy type noise. The sufficient conditions are presented to guarantee the existence of a unique positive solution of the stochastic partial differential equation under investigation. Moreover, we show that positive solutions will blow up in finite time in mean L p -norm sense, provided that the initial data, the nonlinear term and the multiplicative noise satisfies some conditions. Several examples are presented to illustrate the theory. Finally, we establish a global existence theorem based on a Lyapunov functional and prove that a stochastic Allen-Cahn equation driven by Levy noise has a global solution.

Highlights

Fujita [13] considered the initial-boundary problem for a semilinear parabolic equation ∂u ∂t= ∆u + u1+α, t > 0, x ∈ Rd, (1)u(x, 0) = a(x), x ∈ Rd, Fujita showed that there does not exist a global solution for any nontrivial nonnegative initial data when 0 < dα < 2, and there exists a global solution for sufficiently small initial data when dα > 2
U(x, 0) = a(x), x ∈ Rd, Fujita showed that there does not exist a global solution for any nontrivial nonnegative initial data when 0 < dα < 2, and there exists a global solution for sufficiently small initial data when dα > 2
Fujita [14] studied the initial-boundary problem for a semilinear parabolic equation in domain D ⊂ Rd:

Summary

Introduction

Fujita [13] considered the initial-boundary problem for a semilinear parabolic equation. U(x, 0) = a(x), x ∈ Rd, Fujita showed that there does not exist a global solution for any nontrivial nonnegative initial data when 0 < dα < 2, and there exists a global solution for sufficiently small initial data when dα > 2. (f.5) f (r) − λ0r > 0 for r > D a0φ0dx, where a0 = exp(−k|x|2), k > 0, x ∈ Rd. Fujita [14] showed that if D is bounded, a(x) ≥ 0 in D and f satisfies (f.1)(f.5), the solution of (2) blows up in finite time. It is of interest to study the non-existence of global solutions to parabolic stochastic partial differential equations perturbed by random noise as follows: du = ∆u + f (u) + σ(u)dWt, t > 0, x ∈ D, u(x, 0) = g(x), x ∈ D,.

The conclusion is that for

Φr where