Partial differential equations and random fields have been used as successful models in various areas of applied mathematics, statistical mechanics, theoretical physics, theoretical neuroscience, theory of complex chemical reactions, fluid dynamics, hydrology, cosmology, mathematical finance, and other scientific areas. In this talk I will consider non-linear space-time fractional (stochastic) heat type equations. These types of time fractional (stochastic) heat type equations are attractive models that can be used to model phenomenon with random effects with thermal memory.

I will briefly review my most recent work on (i) continuous time random walk limits; (ii) heat type Cauchy problems with fractional time derivatives; and (iii) stochastic fractional equations. In particular, I will talk about the asymptotic behavior of the solution with respect to time and a parameter $\lambda$, and mention some non-existence (blow-up) of global (random field) solutions under some additional conditions.

These results are our recent joint work with Jebessa B Mijena, Mohammud Foondun, Sunday Asogwa and Guerngar Ngartelbaye.

S. Asogwa, J. B. Mijena and E. Nane. Blow-up results for space-time fractional stochastic partial differential equations. Potential Anal., To Appear, 2019. URL:https://arxiv.org/abs/1803.05890.

Z-Q. Chen, Mark M. Meerschaert and E. Nane

M. Foondun, W. Liu, and E. Nane. Some non-existence results for a class of stochastic partial differential equations. J. Differential Equations. Volume 266, Issue 5, 15 February 2019, Pages 2575-2596.

M. Foondun and E. Nane. Asymptotic properties of some space-time fractional stochastic equations. Math. Z. (2017), 1--27. doi:10.1007/s00209-016-1834-3

M. Foondun, J. B. Mijena and E. Nane. Non-linear noise excitation for some space-time fractional stochastic equations in bounded domains. Fract. Calc. Appl. Anal. Vol. 19, No 6 (2016), 1527-1553, DOI: 10.1515/fca-2016--0079.

M.M. Meerschaert, E. Nane and P. Vellaisamy, Fractional Cauchy problems on bounded domains.

Ann. Probab. 37 (2009), 979{1007.

M.M. Meerschaert, E. Nane, and P. Vellaisamy, The fractional Poisson process and the inverse stable subordinator. Elect. J. Probab. 16 (2011), 1600-1620.

M.M. Meerschaert, E. Nane, and P. Vellaisamy, Distributed-order fractional Cauchy problems on bounded domains. J. Math. Anal. Appl. 379 (2011), 216-228.

M.M. Meerschaert, E. Nane, and P. Vellaisamy, Transient anomalous subdiffusions on bounded domains. Proc. Amer. Math. Soc. 141 (2013), 699-710.

M. M. Meerschaert, E. Nane and Y. Xiao, Correlated continuous time random walks. Statist. Probab. Lett. 79 (2009), 1194-1202.

M. M. Meerschaert, E. Nane and Y. Xiao, Fractal dimensions for continuous time random walk limits. Statist. Probab. Lett. 83 (2013), 1083-1093.

J. Mijena and E. Nane. Space time fractional stochastic partial differential equations. Stochastic Process Appl. 125 (2015), no. 9, 3301--3326.% 3301 ñ- 3326

J. B. Mijena, and E.Nane. Intermittence and time fractional partial differential equations. Potential Anal. 44 (2016), 295--312.