## Decay estimates for time-fractional and other non-local in time subdiffusion equations in $$\mathbb R^d$$.(English)Zbl 1354.35178

The authors study the asymptotic behavior of the solution to the equation $\partial_t(k*[u-u_0]) - \Delta u=0, \quad t >0, \quad x\in \mathbb R^d,$ where $$u_{|t=0}= u_0$$ and $$k$$ is locally integrable, non-negative, and non-increasing with $$\lim_{t\downarrow 0} k(t)=+\infty$$. The most important special case that is given most attention is the case of a fractional derivative of order $$\alpha$$, i.e., $$k(t)=\frac 1{\Gamma(1-\alpha)}t^{-\alpha}$$. In addition they study the case where the Laplacian $$\Delta$$ is replaced by a more general elliptic operator in divergence form $$\div(A(t,x)\nabla u)$$ with merely measurable, uniformly elliptic coefficient matrix $$A$$.
For the case of the fractional derivative of order $$\alpha$$ the authors obtain quite precise results, e.g. $$|u(t,\cdot)|_2 \lessapprox t^{-\min\{\frac {\alpha d}4,\alpha\}}$$ when $$d\neq 4$$ and $$|u(t,\cdot)|_{2,\infty} \lessapprox t^{-\alpha}$$ when $$d= 4$$ and thus they establish the existence of a critical dimension $$d=4$$.
The techniques used involve the fundamental solution, Fourier multiplier methods and energy estimates.

### MSC:

 35R11 Fractional partial differential equations 45K05 Integro-partial differential equations 47G20 Integro-differential operators
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### References:

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