In this paper, we consider the two-species predator–prey system with indirect pursuit-evasion interaction and density-dependent motilities {[ u_t=Δ (d_1(w)u)+u(λ -u+av), x∈Ω , t>0,; v_t=Δ (d_2(z)v)+v(μ -v-bu), x∈Ω , t>0,; w_t=d_3Δ w-α _1w+β _1v, x∈Ω , t>0,; z_t=d_4Δ z-α _2z+β _2u, x∈Ω , t>0, ]. subject to homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊆ℝ^2 , where λ , μ , a, b, d_3 , d_4 , α _i , β _i are positive constants for i=1,2 , and the functions d_i(s) satisfy the assumptions: d_i(s) ∈ C^2([0,∞ )) , d_i(s)>0 , d_i^'(s)<0 for s ∈ [0,∞ ) , lim _s→∞d_i^'(s)/d_i(s) is bounded. It is shown that the corresponding initial-boundary value problem admits a global-in-time classical solution in two-dimensional, which is uniformly bounded. Moreover, based on the higher energy estimate and the suitable Lyapunov functionals, we show that (i) If bλ <μ and the parameters satisfy certain conditions, then the solution (u, v, w, z) exponentially converges to (λ +aμ/1+ab, μ -bλ/1+ab, β _1(μ -bλ )/α _1(1+ab), β _2(λ +aμ )/α _2(1+ab)) as t→∞ . (ii) If bλ≥μ and a is large enough, the solution (u, v, w, z) converges to (λ , 0, 0, β _2/α _2λ ) as t→∞ with exponential decay when bλ >μ , and with algebraic decay when bλ =μ .
We consider the problem of proving uniqueness of the solution of the continuity equation with a vector field u ∈ [L^1 (0,T; W^1,p(𝕋^d)) ∩ L^∞ ((0,T) ×𝕋^d)]^d and an initial datum ρ _0 ∈ L^q (𝕋^d) , where 𝕋^d is the d-dimensional torus and 1 ≤ p,q ≤ +∞ such that 1/p + 1/q =1 without using the theory of renormalized solutions as in [6]. We propose a more geometric approach which will, however, still rely on a strong L^1 estimate on the commutator (which ultimately is the key technical tool in [6], too), but other than that will be based on the theory of currents from geometric measure theory.
We investigate an inertial transport–diffusion equation on the half-line with a time-dependent semilinear reaction and Robin boundary condition. The model is formulated in the natural energy space associated with the transformed selfadjoint spatial operator, avoiding the incorrect inference that the second-order matrix generator follows only from the C_0 -semigroup property of the spatial operator. We establish local well-posedness of mild solutions and derive quantitative continuous-dependence estimates. We also identify concrete parameter regimes for the half-line PDE in which the linear semigroup is exponentially stable, and we show how this stability persists under small globally Lipschitz nonlinear perturbations. An exponential weighting restores a symmetric energy framework, yielding balance laws that separate linear damping from nonlinear energy exchange. The admissible nonlinearities include globally Lipschitz and bounded sublinear reactions; genuinely superlinear polynomial terms require additional assumptions beyond the present framework.
We investigate the global existence and large-time behavior of solutions to the non-isentropic Euler equations with damping in ℝ^d ( d ≥ 2 ). By establishing uniform dissipation estimates for the pressure and velocity in both low- and high-frequency regimes, we prove the global well-posedness of classical solutions generated by small initial perturbations in critical hybrid Besov spaces of L^p type. Under an additional assumption on the low-frequency part of the initial data, we further derive sharp convergence rates: the pressure decays optimally as in the heat equation, while the velocity exhibits the optimal decay predicted by Darcy’s law. In addition, we show that the entropy converges to a spatial profile with an explicit rate. To handle the degenerate entropy variable, we employ the localized energy method developed in [40] to get sharp spectrally localized estimates, which allows us to obtain decay for the highest-order derivative and for the critical Lipschitz norm.
This paper investigates the initial periodic-boundary value problem on 𝕋^2 for the non-resistive, heat-conducting compressible magnetohydrodynamics system. By fully exploiting enhanced dissipation mechanisms arising from hidden structures and interactions, particularly the stabilizing effects of the background magnetic field and the additional regularization provided by velocity divergence, we establish the global well-posedness and large-time-decay rates for solutions near a background magnetic field in Sobolev spaces. The absence of dissipation in the density and magnetic field equations poses significant mathematical challenges, as these quantities satisfy forced transport equations. Our analysis employs pure energy estimates, a direct approach that substantially reduces technical complexity. To our knowledge, this represents the first global well-posedness result for this system with precise large-time-decay estimates.
