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 Dirichlet boundary conditions on u. We say that u blows up in finite time, with positive probability, if there is a finite random time T such that P(\lim_{t->T}sup_x u(t,x)=infty)>0. It was known that if gamma<3/2, then with probability 1, u does not blow up in finite time. It was also known that there is a positive probability of finite time blow-up for gamma sufficiently large. In this paper, we show that if gamma>3/2, then there is a positive probability that u blows up in finite time.
State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
Heat kernel estimates for Markov processes with blowing-up jump kernels
University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Seoul National University（首尔国立大学）；Dr. Franjo Tuđman Defense and Security University（弗兰约·图季曼国防与安全大学）；University of Zagreb（萨格勒布大学）
Delayed Blow-up in 3D Fluids via Pseudo-transport Noise
University of Chinese Academy of Sciences（中国科学院大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Università di Pisa（比萨大学）
