Finite-time blowup and existence of global positive solutions of a semi-linear SPDE

Abstract

International audienceWe consider stochastic equations of the prototype $du(t,x) =\left( \Delta u(t,x)+u(t,x)^{1+\beta}\right)dt+\kappa u(t,x)\,dW_{t}$ on a smooth domain $D\subset \mathord{\rm I\mkern-3.6mu R\:\!\!}^d$, with Dirichlet boundary condition, where $\beta$, $\kappa$ are positive constants and $\{W_t $, $t\ge0\}$ is a one-dimensional standard Wiener process. We estimate the probability of finite time blowup of positive solutions, as well as the probability of existence of non-trivial positive global solutions

Exponential Functionals of Brownian Motion and Explosion Times of a System of Semilinear SPDEs

“…In a previous work [2] we investigated blow-up times of semilinear SPDEs of the prototype du(t; x) = u(t; x) + u 1+ (t; x) dt + u(t; x) dW t ; x 2 D; (1.1)…”

“…In case of = 0, the bounds we found give the result of Fujita quoted above. We refer to [2] for de…nitions of blow-up times, and for types of solutions of SPDEs. In [2] it is also shown that the asymptotic behavior of (1.1) is determined to a great extent by the distribution of the exponential functional Z t 0 expf W r ( + 2 =2)rg dr:…”

“…From the results of [2] it follows that for initial values of the form u(0; x) = k (x), x 2 D, where k > 0 is a parameter, the explosion time % of (1.1) satis…es % , where = inf t 0 :…”

“…2 and [13], Proposition 2; see also [8]). The upper bound of % was achieved in [2] by determining the blow-up time of a subsolution of Eq. (1.1), while for the lower bound , a scheme of successive approximations for the mild solution of (1.1) was used.…”

“…These choices make it possible to obtain explicit solutions of a related system of random PDEs, and to use Yor's formula and some extensions of it. Moreover, in contrast with the case treated in [2], the blow-up times of (1.2) are …nite with probability one; see e.g. [7], Prop.…”