Global attractors for the wave equation with nonlinear damping

Based on a new a priori estimate method, so-called asymptotic a priori estimate, the existence of a global attractor is proved for the wave equation utt+kg(ut)−Δu+f(u)=0 on a bounded domain Ω⊂R³ with Dirichlet boundary conditions. The nonlinear damping term g is supposed to satisfy the growth condition C₁(|s|−C₂)?|g(s)|?C₃(1+p|s|), where 1?p<5; the damping parameter is arbitrary; the nonlinear term f is supposed to satisfy the growth condition |f^′(s)|?C₄(1+q|s|), where q?2. It is remarkable that when 2<p<5, we positively answer an open problem in Chueshov and Lasiecka [I. Chueshov, I. Lasiecka, Long-time behavior of second evolution equations with nonlinear damping, Math. Scuola Norm. Sup. (2004)] and improve the corresponding results in Feireisl [E. Feireisl, Global attractors for damped wave equations with supercritical exponent, J. Differential Equations 116 (1995) 431-447].     

Uniqueness of positive radial solutions for quasilinear elliptic equations in an annulus

Extending a previous result of Tang [1] we prove the uniqueness of positive radial solutions of Δ_pu+f(u)=0, subject to Dirichlet boundary conditions on an annulus in Rⁿ with 2<p≤n, under suitable hypotheses on the nonlinearity f. This argument also provides an alternative proof for the uniqueness of positive solutions of the same problem in a finite ball (see [9]), in the complement of a ball or in the whole space Rⁿ (see [10], [3] and [11]).     

Blowup rate for heat equation in Lipschitz domains with nonlinear heat source terms on the boundary

This paper establishes the blowup rate estimate near the blowup time for the heat equation u_t=Δu with the nonlinear boundary condition u_n=u^p on the boundary ∂Ω×[0,T) for a bounded Lipschitz domain and for the nonlinear boundary condition u_n=exp(u) on the boundary for a two-dimensional disc with nonradially symmetric data.     

Convergence to strong nonlinear diffusion waves for solutions to p-system with damping on quadrant

In this paper, we consider the so-called p-system with linear damping on quadrant. We show that for a certain class of given large initial data (v₀(x),u₀(x)), the corresponding initial-boundary value problem admits a unique global smooth solution (v(x,t),u(x,t)) and such a solution tends time-asymptotically, at the Lp (2?p?∞) optimal decay rates, to the corresponding nonlinear diffusion wave which satisfies (1.9) provided the corresponding prescribed initial error function (V₀(x),U₀(x)) lies in (H³(R⁺)∩L¹(R⁺))×(H²(R⁺)∩L¹(R⁺)).     

Removable singularity of the polyharmonic equation

Let x₀∈Ω⊂Rⁿ, n≥2, be a domain and let m≥2. We will prove that a solution u of the polyharmonic equation Δ^mu=0 in Ω?{x₀} has a removable singularity at x₀ if and only if as |x−x₀|→0 for n≥3 and as |x−x₀|→0 for n=2. For m≥2 we will also prove that u has a removable singularity at x₀ if |u(x)|=o(|x−x₀|^2m−n) as |x−x₀|→0 for n≥3 and |u(x)|=o(|x−x₀|^2m−2log(|x−x₀|⁻¹)) as |x−x₀|→0 for n=2.     

Blow-up theorem for semilinear wave equations with non-zero initial position

One of the features of solutions of semilinear wave equations can be found in blow-up results for non-compactly supported data. In spite of finite propagation speed of the linear wave, we have no global in time solution for any power nonlinearity if the spatial decay of the initial data is weak. This was first observed by Asakura (1986) [2] finding out a critical decay to ensure the global existence of the solution. But the blow-up result is available only for zero initial position having positive speed.In this paper the blow-up theorem for non-zero initial position by Uesaka (2009) [22] is extended to higher-dimensional case. And the assumption on the nonlinear term is relaxed to include an example, |u|^p−1u. Moreover the critical decay of the initial position is clarified by example.     

Blowup stability of solutions of the nonlinear heat equation with a large life span

We study the Cauchy problem for the nonlinear heat equation u_t-?u=|u|^p-1u in R^N. The initial data is of the form u₀=λ?, where ?∈C₀(R^N) is fixed and λ>0. We first take 1<p<p_f, where p_f is the Fujita critical exponent, and ?∈C₀(R^N)∩L¹(R^N) with nonzero mean. We show that u(t) blows up for λ small, extending the H. Fujita blowup result for sign-changing solutions. Next, we consider 1<p<p_s, where p_s is the Sobolev critical exponent, and ?(x) decaying as |x|^-σ at infinity, where p<1+2/σ. We also prove that u(t) blows up when λ is small, extending a result of T. Lee and W. Ni. For both cases, the solution enjoys some stable blowup properties. For example, there is single point blowup even if ? is not radial.     

Existence and non-existence of solutions for a class of Monge-Ampère equations

We study the boundary value problems for Monge-Ampère equations: detD²u=e−u in Ω⊂Rn, n?1, u_|∂Ω=0. First we prove that any solution on the ball is radially symmetric by the argument of moving plane. Then we show there exists a critical radius such that if the radius of a ball is smaller than this critical value there exists a solution, and vice versa. Using the comparison between domains we can prove that this phenomenon occurs for every domain. Finally we consider an equivalent problem with a parameter detD²u=e−tu in Ω, u_|∂Ω=0, t?0. By using Lyapunov-Schmidt reduction method we get the local structure of the solutions near a degenerate point; by Leray-Schauder degree theory, a priori estimate and bifurcation theory we get the global structure.     

Second critical exponent and life span for pseudo-parabolic equation

This paper studies the second critical exponent and life span of solutions for the pseudo-parabolic equation u_t−kΔu_t=Δu+u^p in Rⁿ×(0,T), with p>1, k>0. It is proved that the second critical exponent, i.e., the decay order of the initial data required by global solutions in the coexistence region of global and non-global solutions, is independent of the pseudo-parabolic parameter k. Nevertheless, it is revealed that the viscous term kΔu_t relaxes restrictions on the amplitude of the initial data required by the global solutions. Moreover, it is observed that the life span of the non-global solutions will be delayed by the third order viscous term. Finally, some numerical examples are given to illustrate all these results.     

Existence and asymptotic behavior for elliptic equations with singular anisotropic potentials

We study the equation Δu+u|u|^p−1+V(x)u+f(x)=0 in Rn, where n?3 and p>n/(n−2). The forcing term f and the potential V can be singular at zero, change sign and decay polynomially at infinity. We can consider anisotropic potentials of form h(x)|x|⁻² where h is not purely angular. We obtain solutions u which blow up at the origin and do not belong to any Lebesgue space Lr. Also, u is positive and radial, in case f and V are. Asymptotic stability properties of solutions, their behavior near the singularity, and decay are addressed.     

Some estimates of the norm of solutions of nonlinear elliptic eigenvalue problems

We obtain estimates on the possible growth or decay rates as λ → 0 of sup |u^λ|, where uλ ? O satisfies the nonlinear elliptic boundary value problen Lu_λ = λ f(x,u_λ) in a bounded domain subject to homogensous Dirichlet boundary conditions. The estimates generalize existing results by allowing f(x,O) ≠ 0. The analysis is based on integration by parts and Sobolev inequalitie.     

Asymptotics of the porous media equation via Sobolev inequalities

Let M be a compact Riemannian manifold without boundary. Consider the porous media equation , u(0)=u₀∈L^q, ? being the Laplace-Beltrami operator. Then, if q?2∨(m-1), the associated evolution is L^q-L^∞ regularizing at any time t>0 and the bound ‖u(t)‖_∞?C(u₀)/t^β holds for t<1 for suitable explicit C(u₀),γ. For large t it is shown that, for general initial data, u(t) approaches its time-independent mean with quantitative bounds on the rate of convergence. Similar bounds are valid when the manifold is not compact, but u(t) approaches u≡0 with different asymptotics. The case of manifolds with boundary and homogeneous Dirichlet, or Neumann, boundary conditions, is treated as well. The proof stems from a new connection between logarithmic Sobolev inequalities and the contractivity properties of the nonlinear evolutions considered, and is therefore applicable to a more abstract setting.     

Large time behavior and L−L estimate of solutions of 2-dimensional nonlinear damped wave equations

We show the asymptotic behavior of the solution to the Cauchy problem of the two-dimensional damped wave equation. It is shown that the solution of the linear damped wave equation asymptotically decompose into a solution of the heat and wave equations and the difference of those solutions satisfies the L^p−L^q type estimate. This is a two-dimensional generalization of the three-dimensional result due to Nishihara (Math. Z. 244 (2003) 631). To show this, we use the Fourier transform and observe that the evolution operators of the damped wave equation can be approximated by the solutions of the heat and wave equations. By using the L^p−L^q estimate, we also discuss the asymptotic behavior of the semilinear problem of the damped wave equation with the power nonlinearity |u|^αu. Our result covers the whole super critical case α>1, where the α=1 is well known as the Fujita exponent when n=2.     

Oscillatory radial solutions for subcritical biharmonic equations

It is well known that the biharmonic equation Δ²u=u|u|^p−1 with p∈(1,∞) has positive solutions on Rn if and only if the growth of the nonlinearity is critical or supercritical. We close a gap in the existing literature by proving the existence and uniqueness, up to scaling and symmetry, of oscillatory radial solutions on Rn in the subcritical case. Analyzing the nodal properties of these solutions, we also obtain precise information about sign-changing large radial solutions and radial solutions of the Dirichlet problem on a ball.     

Concentrating solutions for a planar elliptic problem involving nonlinearities with large exponent

We consider the boundary value problem Δu+up=0 in a bounded, smooth domain Ω in R² with homogeneous Dirichlet boundary condition and p a large exponent. We find topological conditions on Ω which ensure the existence of a positive solution up concentrating at exactly m points as p→∞. In particular, for a nonsimply connected domain such a solution exists for any given m?1.     

Structure of boundary blow-up solutions for quasi-linear elliptic problems II: small and intermediate solutions

The structure of positive boundary blow-up solutions to quasi-linear elliptic problems of the form −Δ_pu=λf(u), u=∞ on ∂Ω, 1<p<∞, is studied in a bounded smooth domain , for a class of nonlinearities f∈C¹((0,∞)?{z₂})∩C⁰[0,∞) satisfying f(0)=f(z₁)=f(z₂)=0 with 0<z₁<z₂, f<0 in (0,z₁)∪(z₂,∞), f>0 in (z₁,z₂). Large, small and intermediate solutions are obtained for λ sufficiently large. It is known from Part I (see Structure of boundary blow-up solutions for quasilinear elliptic problems, part (I): large and small solutions, preprint), that the large solution is the unique large solution to the problem. We will see that the small solution is also the unique small solution to the problem while there are infinitely many intermediate solutions. Our results are new even for the case p=2.     

Coincidence sets in quasilinear elliptic problems of monostable type

This paper concerns the formation of a coincidence set for the positive solution of the boundary value problem: −εΔpu=uq−1f(a(x)−u) in Ω with u=0 on ∂Ω, where ε is a positive parameter, Δpu=div(|∇u|^p−2∇u), 1<q?p<∞, f(s)∼|s|^θ−1s(s→0) for some θ>0 and a(x) is a positive smooth function satisfying Δpa=0 in Ω with infΩ|∇a|>0. It is proved in this paper that if 0<θ<1 the coincidence set Oε={x∈Ω:uε(x)=a(x)} has a positive measure for small ε and converges to Ω with order O(ε1/p) as ε→0. Moreover, it is also shown that if θ?1, then Oε is empty for any ε>0. The proofs rely on comparison theorems and the energy method for obtaining local comparison functions.     

Nonresonance between the first two eigenvalues for a Steklov problem

In this paper, we study the solvability of the Steklov problem Δ_pu=|u|^p−2u in Ω, on ∂Ω, under assumptions on the asymptotic behaviour of the quotients f(x,s)/|s|^p−2s and pF(x,s)/|s|^p which extends the classical results with Dirichlet boundary conditions that for a.e. x∈∂Ω, the limits at the infinity of these quotients lie between the first two eigenvalues.     

EXISTENCE OF FAST-DECAY GROUND STATE OF P-LAPLACIAN EQUATION

Using the shooting argument and an approximating method, this paper is concerned with the existence of fast-decay ground state of p-Laplacian equation: Δ_pu + f(u) = 0, in Rⁿ, where f(u) behaves just like f(u) = u^q – u^s, as s > q >np/(n – p) – 1.