Abstract
Abstract This paper deals with the abstract evolution equations in L s {L}^{s} -spaces with critical temporal weights. First, embedding and interpolation properties of the critical L s {L}^{s} -spaces with different exponents s s are investigated, then solvability of the linear evolution equation, attached to which the inhomogeneous term f f and its average Φ f \Phi f both lie in an L 1 / s s {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 → 0 t\to 0 for some bounded function α ( t ) \alpha \left(t) like ( − log t ) − p {\left(-\log t)}^{-p} with 2 ≤ p < ∞ 2\le p\lt \infty .
Highlights
This paper deals with the abstract evolution equations in Ls-spaces with critical temporal weights or equivalently critical Ls-spaces
Under the assumption Φf ∈ L1s/s(0, T; X), we can conclude that t−1/su ∈ Ls(0, T; X), and ∥u∥L∗s(0,T;X) ≤ M0 ∥Φf ∥L1s/s(0,T)
We can verify that u is exactly the strong solution of (1.6) by checking the inclusion F(⋅, u(⋅)) ∈ L1(0, T; X) ∩ L1∞/∞(0, T; X)
Summary
This paper deals with the abstract evolution equations in Ls-spaces with critical temporal weights or equivalently critical Ls-spaces. Given a Banach space X and a number 0 < T < ∞, let L0(0, T; X) be the collection of strongly measurable X-valued functions. Given two indices 0 < μ ≤ 1 and 1 < s < ∞, the weighted abstract-valued Ls-space and Wk,s-space are defined as follows: Lμs(0, T; X) = {f ∈ L0(0, T; X) : t1−μ∥f (t)∥X ∈ Ls(0, T)}. Wμk,s(0, T; X) = {f ∈ Wl1o,c1(0, T; X) : f (j) (t) ∈ Lμs(0, T; X), j = 0, 1,..., k}
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