Solvability of the abstract evolution equations in Ls-spaces with critical temporal weights

Abstract

This paper deals with the abstract evolution equations in Ls{L}^{s}-spaces with critical temporal weights. First, embedding and interpolation properties of the critical Ls{L}^{s}-spaces with different exponents ss are investigated, then solvability of the linear evolution equation, attached to which the inhomogeneous term ff and its average Φf\Phi f both lie in an L1/ss{L}_{1\hspace{-0.08em}\text{/}\hspace{-0.08em}s}^{s}-space, is established. Based on these results, Cauchy problem of the semi-linear evolution equation is treated, where the nonlinear operator F(t,u)F\left(t,u) has a growth number ρ≥s+1\rho \ge s+1, and its asymptotic behavior acts like α(t)/t\alpha \left(t)\hspace{-0.1em}\text{/}\hspace{-0.1em}t as t→0t\to 0 for some bounded function α(t)\alpha \left(t) like (−logt)−p{\left(-\log t)}^{-p} with 2≤p<∞2\le p\lt \infty .

Keywords

• critical l s-space
• sectorial operator
• maximal regularity
• evolution equation
• solvability
• 34k30
• 47d06