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 .

中文翻译：

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具有临界时间权重的Ls空间中抽象演化方程的可解性

本文讨论具有临界时间权重的L s {L} ^ {s}-空间中的抽象演化方程。首先，研究具有不同指数ss的临界L s {L} ^ {s}-空间的嵌入和内插特性，然后研究线性演化方程的可解性，该方程具有非齐次项ff及其平均Φf \ Phi f两者都位于L 1 / ss {L} _ {1 \ hspace {-0.08em} \ text {/} \ hspace {-0.08em} s} ^ {s} -space中。基于这些结果，处理了半线性发展方程的柯西问题，其中非线性算子F（t，u）F \ left（t，u）的增长数为ρ≥s + 1 \ rho \ ge s + 1，其渐近行为的行为类似于α（t）/ t \ alpha \ left（t）\ hspace {-0.1em} \ text {/} \ hspace {-0。