Narrow Results

In this paper, we consider a diffusion equation coupled to an ordinary differential equation, modeling the interaction between receptors and ligands. We first prove the existence of global-in-time solutions for all bounded and non-negative initial conditions. Then large-time behavior of solutions is studied. Our result shows that each positive solution converges to the constant solutions as time $t$ tends to infinity. Finally, we construct both traveling back and traveling front solutions by applying the blow up method.

We discuss the time evolution of the wave function which is the solution of a stochastic Schrödinger equation describing the dynamics of a free quantum particle subject to spontaneous localizations in space. We prove global existence and uniqueness of solutions. We observe that there exist three time regimes: the collapse regime, the classical regime and the diffusive regime. Concerning the latter, we assert that the general solution converges almost surely to a diffusing Gaussian wave function having a finite spread both in position as well as in momentum. This paper corrects and completes earlier works on this issue.

In this paper, we study the Cauchy problem for one dimension generalized damped Boussinesq equation. First, global existence and decay estimate of solutions to this problem are established. Second, according to the detail analysis for solution operator the generalized damped Boussinesq equation, the nonlinear approximation to global solutions is established. Finally, we prove that the global solution u to our problem is asymptotic to the superposition of nonlinear diffusion waves expressed in terms of the self-similar solution of the viscous Burgers equation as time tends to infinity.

This work deals with a taxis cascade model for food consumption in two populations, namely foragers directly orienting their movement upward the gradients of food concentration and exploiters taking a parasitic strategy in search of food via tracking higher forager densities. As a consequence, the dynamics of both populations are adapted to the space distribution of food which is dynamically modified in time and space by the two populations. This model extends the classical one-species chemotaxis-consumption systems by additionally accounting for a second taxis mechanism coupled to the first in a consecutive manner. It is rigorously proved that for all suitably regular initial data, an associated Neumann-type initial-boundary value problem for the spatially one-dimensional version of this model possesses a globally defined bounded classical solution. Moreover, it is asserted that the considered two populations will approach spatially homogeneous distributions in the large time limit, provided that either the total population number of foragers or that of exploiters is appropriately small.

As a simplified version of a three-component taxis cascade model accounting for different migration strategies of two population groups in search of food, a two-component nonlocal nutrient taxis system is considered in a two-dimensional bounded convex domain with smooth boundary. For any given conveniently regular and biologically meaningful initial data, smallness conditions on the prescribed resource growth and on the initial nutrient signal concentration are identified which ensure the global existence of a global classical solution to the corresponding no-flux initial-boundary value problem. Moreover, under additional assumptions on the food production source these solutions are shown to be bounded, and to stabilize toward semi-trivial equilibria in the large time limit, respectively.

This paper is concerned with the higher-dimensional haptotactic system modeling oncolytic virotherapy, which was initially proposed by Alzahrani–Eftimie–Trucu [Multiscale modelling of cancer response to oncolytic viral therapy, Math. Biosci. 310 (2019) 76–95] (see also the survey Bellomo–Outada et al. [Chemotaxis and cross-diffusion models in complex environments: Models and analytic problems toward a multiscale vision, Math. Models Methods Appl. Sci. 32 (2022) 713–792]) to model the process of oncolytic viral therapy. We consider this problem in a bounded domain $\mathrm{\Omega }\subseteq {ℝ}^N(N=2,3)$ with zero-flux boundary conditions. Although the $L^{\infty }$-norm of the extracellular matrix density $v$ is easily obtainable, the remodeling process still causes difficulty due to the deficiency of regularity for $v$. Relying on some $L^p$-estimate techniques, in this paper, under the mild condition on parameters, we finally established the existence of global-in-time classical solution, which is bounded uniformly. Moreover, the large time behavior of solutions to the problem is also investigated. Specially speaking, when ${\kappa }_u=0$, the corresponding solution of the system decays to $(0,0,0)$ algebraically. To the best of our knowledge, these are the first results on boundedness and asymptotic behavior of the system in three-dimensional space.

This paper studies the global existence and the asymptotic self-similar behavior of solutions of the semilinear parabolic system ∂_tu=Δu+a₁|u|^p₁-1u+b₁|v|^q₁-1v, ∂_tv=Δv+a₂|v|^p₂-1v+b₂|u|^q₂-1u, on (0,∞)×ℝⁿ, where a₁, b_i∈ℝ and p_i, q_i>1. Let p=min{p₁,p₂, q₁(1+q₂)/(1+q₁), q₂(1+q₁)/(1+q₂)}. Under the condition p>1+2/n we prove the existence of globally decaying mild solutions with small initial data. Some of them are asymptotic, for large time, to self-similar solutions of appropriate asymptotic systems having each one a self-similar structure. All possible asymptotic self-similar behaviors are discussed in terms of exponents p_i, q_i, the space dimension n and the asymptotic spatial profile of the related initial data.

We study the dynamics of the following porous medium equation with strong absorption

In this work, we show the spectrum structure of the linear Fokker–Planck equation by using the semigroup theory and the linear operator perturbation theory. As an application, we show the large time behavior of the solutions to the linear Fokker–Planck equation.

We investigate an initial boundary value problem of two-dimensional nonhomogeneous heat conducting magnetohydrodynamic equations. We prove that there exists a unique global strong solution. Moreover, we also obtain the large time decay rates of the solution. Note that the initial data can be arbitrarily large and the initial density allows vacuum states. Our method relies upon the delicate energy estimates and Desjardins’ interpolation inequality (B. Desjardins, Regularity results for two-dimensional flows of multiphase viscous fluids, Arch. Rational Mech. Anal.137(2) (1997) 135–158).

We study the large time behavior of entropic approximate solutions to one-dimensional, hyperbolic conservation laws with periodic initial data. Under mild assumptions on the numerical scheme, we prove the asymptotic convergence of the discrete solutions to a time- and space-periodic solution.

The Euler–Maxwell system regarded as a hydrodynamic model for plasma physics describes the dynamics of 'compressible electrons' in a constant, charged, non-moving ion background. A global smooth flow with small amplitude is constructed here in three space dimensions when the electron velocity relaxation is taken into account. The speed of the electron flow tending to a uniform equilibrium, and the pointwise behavior of solutions to the linearized homogeneous system in the frequency space are investigated in detail.

We study the large time behavior of solutions to the space-time monopole equations in $1+1$ dimensions. We establish that the solutions will tend to traveling wave solution when time tends to infinity.