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Cauchy problem for fractional diffusion equations. (English) Zbl 1068.35037
Equations of the form $$(D^{(\alpha)}_tu)(t,x)-B u(t,x)=f(t,x),\quad t\in[0,\tau], \quad 0<\alpha<1,\ x\in\Bbb R^n$$ where $$(D^{(\alpha)}_tu)(t,x)= \frac1{\Gamma(1-2)}\left[\frac\partial{\partial t}\int^t_0(t-\zeta)^{-\alpha}u(\zeta,x)\,d\zeta-t^{-\alpha}u(0,x)\right]$$ $$B= \sum^n_{k,j=1} a_{ij}(x)\frac{\partial^2}{\partial x_i\partial x_j}+\sum^n_{j=1}b_j(x)\frac{\partial}{\partial x_j}+c(x)$$ are considered here. The fundamental solution is studied via a Green matrix. The arguments of the Green matrix are expresssed in terms of Fox’s $H$-functions. Estimates of the elements of the Green matrix are also presented.

##### MSC:
 35K15 Second order parabolic equations, initial value problems 26A33 Fractional derivatives and integrals (real functions)
##### Keywords:
Fox’s $H$-function; fundamental soloution
Full Text:
##### References:
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