Abstract The goal of this paper is to obtain estimates for nonnegative solutions of the differential inequality $$\begin{aligned} \left( \frac{\partial }{\partial t} - \Delta \right) u \le A u^p + B u \nonumber \end{aligned}$$ ∂ ∂ t - Δ u ≤ A u p + B u with small initial data in borderline Morrey norms over a Riemannian manifold with bounded geometry. We obtain $$L^\infty $$ L ∞ estimates assuming $$\Vert u(\cdot ,0)\Vert _{M^{q, \frac{2q}{p-1}}} + \sup _{0 \le t< T} \Vert u(\cdot , t) \Vert _{L^s} < \delta ,$$ ‖ u ( · , 0 ) ‖ M q , 2 q p - 1 + sup 0 ≤ t < T ‖ u ( · , t ) ‖ L s < δ , where $$1 < q \le q_c:= \frac{n(p-1)}{2}$$ 1 < q ≤ q c : = n ( p - 1 ) 2 and $$1 \le s \le q_c$$ 1 ≤ s ≤ q c . Assuming also a bound on $$\Vert u(\cdot , 0)\Vert _{M^{q', \lambda '}}$$ ‖ u ( · , 0 ) ‖ M q ′ , λ ′ , where $$\frac{\lambda '}{2q'} < \frac{1}{p-1},$$ λ ′ 2 q ′ < 1 p - 1 , we get an improved estimate near the initial time. These results have applications to geometric flows in higher dimensions.

Abstract This paper is concerned with a diffusive Lotka–Volterra cooperative model with population flux by attractive transition. We study the global well-posedness and the large time behavior of solutions in a case where the habitat is a bounded convex domain and random diffusion rates equal to each other. The main result shows that when the spatial dimension is less than or equal to 3, under a weaker cooperative condition, a classical solution exists globally in time if the initial data belong to a suitable functional space. Furthermore, it is shown that if there exists a positive steady state and the equal random diffusion rate is sufficiently large, then all positive solutions asymptotically approach the positive steady state as time tends to infinity.

This paper is concerned with the L1 in time Bq,1s in space maximal regularity for the Stokes equations obtained by linearization procedure of the Navier–Stokes equations describing the viscous compressible fluid motion. Our main tool of deriving this maximal regularity is based on the spectral analysis of the corresponding resolvent problem for the Stokes operators. An application of our theorem is to prove the local well-posedness of the Navier–Stokes equations with non-slip boundary conditions in uniform C3 domains, whose boundary is compact. This is an extension of results due to Danchin and Tolksdorf (Math Ann 387:1903–1959, 2023), where the boundedness of the domain is assumed. In this paper, we assume that the boundary of the domain is compact, namely not only bounded domains but also exterior domains are considered. Our approach of this paper is based on the spectral analysis of Lamé equations, while the method in Danchin and Tolksdorf (Math Ann 387:1903–1959, 2023) is an extension of a result due to Da Prato and Grisvard (J Math Pures Appl (9) 54(3):305–387, 1974). Our method developed in this paper has applications to extensive system of parabolic and hyperbolic–parabolic equations with non-homogeneous boundary conditions.