Blow up of the nonclassical diffusion equation

2009 ◽  
Vol 50 (4) ◽  
pp. 042702 ◽  
Author(s):  
Haitao Song ◽  
Chengkui Zhong
Keyword(s):  
Diffusion Equation ◽  
Blow Up ◽  
Nonclassical Diffusion Equation

On a stochastic nonclassical diffusion equation with standard and fractional Brownian motion

2021 ◽  
pp. 2140011
Author(s):  
Tomás Caraballo ◽  
Tran Bao Ngoc ◽  
Tran Ngoc Thach ◽  
Nguyen Huy Tuan
Keyword(s):  
Brownian Motion ◽  
Diffusion Equation ◽  
Stochastic Integrals ◽  
Well Posedness ◽  
Time Data ◽  
Final Time ◽  
Fractional Brownian Motions ◽  
Nonclassical Diffusion Equation ◽  
Definition Of ◽  
Well Posed

This paper is concerned with the mathematical analysis of terminal value problems (TVP) for a stochastic nonclassical diffusion equation, where the source is assumed to be driven by classical and fractional Brownian motions (fBms). Our two problems are to study in the sense of well-posedness and ill-posedness meanings. Here, a TVP is a problem of determining the statistical properties of the initial data from the final time data. In the case [Formula: see text], where [Formula: see text] is the fractional order of a Laplace operator, we show that these are well-posed under certain assumptions. We state a definition of ill-posedness and obtain the ill-posedness results for the problems when [Formula: see text]. The major analysis tools in this paper are based on properties of stochastic integrals with respect to the fBm.

On the quenching set for a fast diffusion equation: regional quenching

2005 ◽  
Vol 135 (3) ◽  
pp. 585-602
Author(s):  
R. Ferreira ◽  
A. de Pablo ◽  
F. Quirós ◽  
J. D. Rossi
Keyword(s):  
Boundary Condition ◽  
Diffusion Equation ◽  
Blow Up ◽  
Single Point ◽  
Fast Diffusion ◽  
Fast Diffusion Equation ◽  
Flux Boundary Condition ◽  
Bounded Interval ◽  
Quenching Set ◽  
Type Boundary

We study positive solutions of a very fast diffusion equation, ut = (um−1ux)x, m < 0, in a bounded interval, 0 < x < L, with a quenching-type boundary condition at one end, u (0, t) = (T − t)1/(1 − m) and a zero-flux boundary condition at the other, (um −1ux)(L, t) = 0. We prove that for m ≥ −1 regional quenching is not possible: the quenching set is either a single point or the whole interval. Conversely, if m < −1 single-point quenching is impossible, and quenching is either regional or global. For some lengths the above facts depend on the initial data. The results are obtained by studying the corresponding blow-up problem for the variable v = um −1.

Uniform Blow-Up Rate for Diffusion Equation with a Weighted Localized Source

2013 ◽  
Vol 709 ◽  
pp. 523-526 ◽  
Author(s):  
Miao Chao Chen
Keyword(s):  
Dirichlet Problem ◽  
Nonlinear Equation ◽  
Diffusion Equation ◽  
Blow Up ◽  
Localized Source ◽  
Blow Up Rate ◽  
Compact Subsets

In this paper,we investigate the Dirichlet problem for a localized nonlinear equation with a weighted localized source term.Under certain conditions, the solutions have a global blow-up and the rate of blow-up is uniform in all compact subsets of the domain.Moreover,we obtain the blow-up rate on maximal value point under some appropriate hypotheses.

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