Decay estimates for time-fractional and other non-local in time subdiffusion equations in \(\mathbb R^d\) (Q343156)

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.