Optimal lower bound of the grow-up rate for a supercritical parabolic equation

We study the behavior of solutions of the Cauchy problem for a supercritical semilinear parabolic equation which approach a singular steady state from below as t→∞. It is known that the grow-up rate of such solutions depends on the spatial decay rate of initial data. We give an optimal lower bound on the grow-up rate by using a comparison technique based on a formal asymptotic analysis.     

Young measure solutions and instability of the one-dimensional Perona-Malik equation

We apply a variational approach to the one-dimensional version of the widely used Perona-Malik equation in image processing. We rephrase the problem into the one related to the quasiconvex hull of a graph in the space of 2×2 matrices M2×2. We then use the solutions of some heat equations as the centre of the mass for the Young measure-valued solutions to construct the approximate solutions by using simple laminates. The approximate solutions can be viewed as solutions of a perturbation problem by W−1,p (or W−1,∞) functions. The sequences of the approximate solutions generates Young measure-valued solutions. Our results also show that the solutions of the one-dimensional Perona-Malik equation are unstable under small W−1,∞ perturbations.     

Asymptotic energy concentration in the phase space of the weak solutions to the Navier-Stokes equations

We study the asymptotic behavior of the energy of weak solutions of Navier-Stokes equations as t→∞. We characterize the space of the initial data which causes a concentration of the kinetic energy in the phase space. Moreover, an explicit convergence rate is obtained.     

Locating the peaks of least-energy solutions to a quasilinear elliptic Neumann problem

In this paper we study the shape of least-energy solutions to the quasilinear problem εmΔmu−um−1+f(u)=0 with homogeneous Neumann boundary condition. We use an intrinsic variation method to show that as ε→⁰⁺, the global maximum point Pε of least-energy solutions goes to a point on the boundary ∂Ω at the rate of o(ε) and this point on the boundary approaches to a point where the mean curvature of ∂Ω achieves its maximum. We also give a complete proof of exponential decay of least-energy solutions.     

Existence and decay of solutions in full space to Navier-Stokes equations with delays

We consider the Navier-Stokes equations with delays in Rⁿ,2≤n≤4. We prove existence of weak solutions when the external forces contain some hereditary characteristics and uniqueness when n=2. Moreover, if the external forces satisfy a time decay condition we show that the solution decays at an algebraic rate.     

Gradient blowup rate for a viscous Hamilton–Jacobi equation with degenerate diffusion

This paper is concerned with the gradient blowup rate for the one-dimensional p-Laplacian parabolic equation ${u_t=(|u_x|^{p-2} u_x)_x +|u_x|^q}$ with q > p > 2, for which the spatial derivative of solutions becomes unbounded in finite time while the solutions themselves remain bounded. We establish the blowup rate estimates of lower and upper bounds and show that in this case the blowup rate does not match the self-similar one.     

Boundary blow-up solutions in the unit ball: Asymptotics, uniqueness and symmetry

We calculate the full asymptotic expansion of boundary blow-up solutions (see Eq. (1) below), for any nonlinearity f. Our approach enables us to state sharp qualitative results regarding uniqueness and radial symmetry of solutions, as well as a characterization of nonlinearities for which the blow-up rate is universal. Lastly, we study in more detail the standard nonlinearities f(u)=up, p>1.     

Stability of Riemann solutions with large oscillation for the relativistic Euler equations

We are concerned with entropy solutions of the 2×2 relativistic Euler equations for perfect fluids in special relativity. We establish the uniqueness of Riemann solutions in the class of entropy solutions in L^∞∩BV_loc with arbitrarily large oscillation. Our proof for solutions with large oscillation is based on a detailed analysis of global behavior of shock curves in the phase space and on special features of centered rarefaction waves in the physical plane for this system. The uniqueness result does not require specific reference to any particular method for constructing the entropy solutions. Then the uniqueness of Riemann solutions yields their inviscid large-time stability under arbitrarily largeL¹∩L^∞∩BV_loc perturbation of the Riemann initial data, as long as the corresponding solutions are in L^∞ and have local bounded total variation that allows the linear growth in time. We also extend our approach to deal with the uniqueness and stability of Riemann solutions containing vacuum in the class of entropy solutions in L^∞ with arbitrarily large oscillation.     

Decay estimates for a thermoelastic system in waveguides

We investigate the linear system of thermoelasticity, consisting of an elasticity equation and a heat conduction equation, in a waveguide Ω=(0,1)×Rn−1, with certain boundary conditions. We consider the cases of homogeneous and inhomogeneous systems and prove decay estimates of the solutions, which are a key ingredient to showing the global existence of solutions to non-linear thermoelasticity, after having decomposed the solutions into various parts. We also give a simplified proof to the representation of the solutions to the Cauchy problem of thermoelasticity.     

Global existence, asymptotic behavior and blow-up of solutions for coupled Klein-Gordon equations with damping terms

This paper studies the Cauchy problem for the coupled system of nonlinear Klein-Gordon equations with damping terms. We first state the existence of standing wave with ground state, based on which we prove a sharp criteria for global existence and blow-up of solutions when E(0)<d. We then introduce a family of potential wells and discuss the invariant sets and vacuum isolating behavior of solutions for 0<E(0)<d and E(0)≤0, respectively. Furthermore, we prove the global existence and asymptotic behavior of solutions for the case of potential well family with 0<E(0)<d. Finally, a blow-up result for solutions with arbitrarily positive initial energy is obtained.     

Bifurcation curves of a logistic equation when the linear growth rate crosses a second eigenvalue

We construct the global bifurcation curves, solutions versus level of harvesting, for the steady states of a diffusive logistic equation on a bounded domain, under Dirichlet boundary conditions and other appropriate hypotheses, when a, the linear growth rate of the population, is below λ₂+δ. Here λ₂ is the second eigenvalue of the Dirichlet Laplacian on the domain and δ>0. Such curves have been obtained before, but only for a in a right neighborhood of the first eigenvalue. Our analysis provides the exact number of solutions of the equation for a≤λ₂ and new information on the number of solutions for a>λ₂.     

Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a prey refuge

We study a predator-prey model with Holling type II functional response incorporating a prey refuge under homogeneous Neumann boundary condition. We show the existence and non-existence of non-constant positive steady-state solutions depending on the constant m∈(0,1], which provides a condition for protecting (1−m)u of prey u from predation. Moreover, we investigate the asymptotic behavior of spacially inhomogeneous solutions and the local existence of periodic solutions.     

Local smoothing effects, positivity, and Harnack inequalities for the fast p-Laplacian equation

We study qualitative and quantitative properties of local weak solutions of the fast p-Laplacian equation, t_∂u=Δpu, with 1<p<2. Our main results are quantitative positivity and boundedness estimates for locally defined solutions in domains of Rn×[0,T]. We combine these lower and upper bounds in different forms of intrinsic Harnack inequalities, which are new in the very fast diffusion range, that is when 1<p?2n/(n+1). The boundedness results may be also extended to the limit case p=1, while the positivity estimates cannot.We prove the existence as well as sharp asymptotic estimates for the so-called large solutions for any 1<p<2, and point out their main properties.We also prove a new local energy inequality for suitable norms of the gradients of the solutions. As a consequence, we prove that bounded local weak solutions are indeed local strong solutions, more precisely .     

Locating the peaks of least-energy solutions to a quasilinear elliptic Neumann problem, Part II

We continue our work (Y. Li, C. Zhao, Locating the peaks of least-energy solutions to a quasilinear elliptic Neumann problem, J. Math. Anal. Appl. 336 (2007) 1368-1383) to study the shape of least-energy solutions to the quasilinear problem ε^mΔ_mu−u^m−1+f(u)=0 with homogeneous Neumann boundary condition. In this paper we focus on the case 1<m<2 as a complement to our previous work on the case m≥2. We use an intrinsic variation method to show that as the case m≥2, when ε→0⁺, the global maximum point P_ε of least-energy solutions goes to a point on the boundary ∂Ω at a rate of o(ε) and this point on the boundary approaches a global maximum point of mean curvature of ∂Ω.     

Convergence and decay rate to equilibrium of bounded solutions of quasilinear parabolic equations

We study the convergence and decay rate to equilibrium of bounded solutions of the quasilinear parabolic equation

ut−diva(x,∇u)+f(x,u)=0     

Existence and regularity of extremal solutions for a mean-curvature equation

We study a class of mean curvature equations −Mu=H+λup where M denotes the mean curvature operator and for p?1. We show that there exists an extremal parameter λ^∗ such that this equation admits a minimal weak solutions for all λ∈[0,λ^∗], while no weak solutions exists for λ>λ^∗ (weak solutions will be defined as critical points of a suitable functional). In the radially symmetric case, we then show that minimal weak solutions are classical solutions for all λ∈[0,λ^∗] and that another branch of classical solutions exists in a neighborhood (λ_∗−η,λ^∗) of λ^∗.     

Global solutions and self-similar solutions of semilinear parabolic equations with nonlinear gradient terms

In this paper we consider the Cauchy problem of semilinear parabolic equations with nonlinear gradient terms a(x)|u|^q−1u|∇u|^p. We prove the existence of global solutions and self-similar solutions for small initial data. Moreover, for a class of initial data we show that the global solutions behave asymptotically like self-similar solutions as t→∞.     

Characterization of non-autonomous attractors of a perturbed infinite-dimensional gradient system

In this paper we determine the exact structure of the pullback attractors in non-autonomous problems that are perturbations of autonomous gradient systems with attractors that are the union of the unstable manifolds of a finite set of hyperbolic equilibria. We show that the pullback attractors of the perturbed systems inherit this structure, and are given as the union of the unstable manifolds of a set of hyperbolic global solutions which are the non-autonomous analogues of the hyperbolic equilibria. We also prove, again parallel to the autonomous case, that all solutions converge as t→+∞ to one of these hyperbolic global solutions. We then show how to apply these results to systems that are asymptotically autonomous as t→−∞ and as t→+∞, and use these relatively simple test cases to illustrate a discussion of possible definitions of a forwards attractor in the non-autonomous case.     

Long range scattering for the modified Schrödinger map in two space dimensions

We study the asymptotic behaviour in time of solutions and the theory of scattering for the modified Schrödinger map in two space dimensions. We solve the Cauchy problem with large finite initial time, up to infinity in time, and we determine the asymptotic behaviour in time of the solutions thereby obtained. As a by product, we obtain global existence for small data in Hk∩FHk with k>1. We also solve the Cauchy problem with infinite initial time, namely we construct solutions defined in a neighborhood of infinity in time, with prescribed asymptotic behaviour of the previous type.     

Decay rate of quasilinear wave equation with viscosity

This paper is concerned with the decay rate of solutions for a quasilinear wave equation with viscosity. We use a so-called energy perturbation method to establish decay rate of solutions in terms of energy norm for a class of nonlinear functions. With the help of a comparison lemma of differential inequalities, we establish a relationship between decay rate of solutions and f .