A note on asymptotic behavior of solutions for the one-dimensional bipolar Euler-Poisson system

In this note, we consider a one-dimensional bipolar Euler-Poisson system (hydrodynamic model). This system takes the form of Euler-Poisson with electric field and frictional damping added to the momentum equations. When n₊≠n₋, paper [I. Gasser, L. Hsiao, H.-L. Li, Large time behavior of solutions of the bipolar hydrodynamical model for semiconductors, J. Differential Equations 192 (2003) 326-359] discussed the asymptotic behavior of small smooth solutions to the Cauchy problem of the one-dimensional bipolar Euler-Poisson system. Subsequent to [I. Gasser, L. Hsiao, H.-L. Li, Large time behavior of solutions of the bipolar hydrodynamical model for semiconductors, J. Differential Equations 192 (2003) 326-359], we investigate the asymptotic behavior of solutions to the Cauchy problem with , and obtain the optimal convergence rate toward the constant state . We accomplish the proofs by energy estimates and the decay rates of fundamental solutions of the heat-type equations.     

Stability of travelling waves in a model for conical flames in two space dimensions

This paper deals with the question of the stability of conical-shaped solutions of a class of reaction-diffusion equations in . One first proves the existence of travelling waves solutions with conical-shaped level sets, generalizing earlier results by Bonnet, Hamel and Monneau [SIAM J. Math. Anal. 31 (1999) 80-118; Comm. Partial Differential Equations 25 (2000) 769-819]. One then gives a characterization of the global attractor of these semilinear parabolic equations under some conical asymptotic conditions. Lastly, the global stability of the travelling waves solutions is proved.     

Optimal uniqueness theorems and exact blow-up rates of large solutions

In this paper, we prove some optimal uniqueness results for large solutions of a canonical class of semilinear equations under minimal regularity conditions on the weight function in front of the non-linearity and combine these results with the localization method introduced in [López-Gómez, The boundary blow-up rate of large solutions, J. Differential Equations 195 (2003) 25-45] to prove that any large solution L of Δu=a(x)u^p, p>1, a>0, must satisfy

     

Bubble towers for supercritical semilinear elliptic equations

We construct positive solutions of the semilinear elliptic problem with Dirichet boundary conditions, in a bounded smooth domain Ω⊂R^N(N?4), when the exponent p is supercritical and close enough to and the parameter λ∈R is small enough. As , the solutions have multiple blow up at finitely many points which are the critical points of a function whose definition involves Green's function. Our result extends the result of Del Pino et al. (J. Differential Equations 193(2) (2003) 280) when Ω is a ball and the solutions are radially symmetric.     

Positive entire solutions of semilinear elliptic equations with quadratically vanishing coefficient

We establish that for n?3 and p>1, the elliptic equation Δu+K(x)up=0 in Rn possesses a continuum of positive entire solutions with logarithmic decay at ∞, provided that a locally Hölder continuous function K?0 in Rn?{0}, satisfies K(x)=O(σ|x|) at x=0 for some σ>−2, and ²|x|K(x)=c+O([log|x|]^−θ) near ∞ for some constants c>0 and θ>1. The continuum contains at least countably many solutions among which any two do not intersect. This is an affirmative answer to an open question raised in [S. Bae, T.K. Chang, On a class of semilinear elliptic equations in Rn, J. Differential Equations 185 (2002) 225-250]. The crucial observation is that in the radial case of K(r)=K(|x|), two fundamental weights, and , appear in analyzing the asymptotic behavior of solutions.     

A computer-assisted existence and multiplicity proof for travelling waves in a nonlinearly supported beam

For a nonlinear beam equation with exponential nonlinearity, we prove existence of at least 36 travelling wave solutions for the specific wave speed c=1.3. This complements the result in [Smets, van den Berg, Homoclinic solutions for Swift-Hohenberg and suspension bridge type equations, J. Differential Equations 184 (2002) 78-96.] stating that for almost all there exists at least one solution. Our proof makes heavy use of computer assistance: starting from numerical approximations, we use a fixed point argument to prove existence of solutions “close to” the computed approximations.     

A multiplicity theorem for problems with the p-Laplacian

We consider a nonlinear elliptic problem driven by the p-Laplacian, with a parameter λ∈R and a nonlinearity exhibiting a superlinear behavior both at zero and at infinity. We show that if the parameter λ is bigger than λ₂=the second eigenvalue of , then the problem has at least three nontrivial solutions. Our approach combines the method of upper-lower solutions with variational techniques involving the Second Deformation Theorem. The multiplicity result that we prove extends an earlier semilinear (i.e. p=2) result due to Struwe [M. Struwe, Variational Methods, Springer-Verlag, Berlin, 1990].     

Fokker-Planck equations and maximal dissipativity for Kolmogorov operators with time dependent singular drifts in Hilbert spaces

We consider a Kolmogorov operator L₀ in a Hilbert space H, related to a stochastic PDE with a time-dependent singular quasi-dissipative drift , defined on a suitable space of regular functions. We show that L₀ is essentially m-dissipative in the space Lp([0,T]×H;ν), p?1, where and the family (νt)_t∈[0,T] is a solution of the Fokker-Planck equation given by L₀. As a consequence, the closure of L₀ generates a Markov C₀-semigroup. We also prove uniqueness of solutions to the Fokker-Planck equation for singular drifts F. Applications to reaction-diffusion equations with time-dependent reaction term are presented. This result is a generalization of the finite-dimensional case considered in [V. Bogachev, G. Da Prato, M. Röckner, Existence of solutions to weak parabolic equations for measures, Proc. London Math. Soc. (3) 88 (2004) 753-774], [V. Bogachev, G. Da Prato, M. Röckner, On parabolic equations for measures, Comm. Partial Differential Equations 33 (3) (2008) 397-418], and [V. Bogachev, G. Da Prato, M. Röckner, W. Stannat, Uniqueness of solutions to weak parabolic equations for measures, Bull. London Math. Soc. 39 (2007) 631-640] to infinite dimensions.     

Global structure instability of Riemann solutions of quasilinear hyperbolic systems of conservation laws: Rarefaction waves

This work is a continuation of our previous work (Kong, J. Differential Equations 188 (2003) 242-271) “Global structure stability of Riemann solutions of quasilinear hyperbolic systems of conservation laws: shocks and contact discontinuities”. In the present paper we prove the global structure instability of the Lax's Riemann solution , containing rarefaction waves, of general n×n quasilinear hyperbolic system of conservation laws. Combining the results in (Kong, 2003), we prove that the Lax's Riemann solution of general n×n quasilinear hyperbolic system of conservation laws is globally structurally stable if and only if it contains only non-degenerate shocks and contact discontinuities, but no rarefaction waves and other weak discontinuities.     

Boundary singularities for weak solutions of semilinear elliptic problems

Let Ω be a bounded domain in RN, N?2, with smooth boundary ∂Ω. We construct positive weak solutions of the problem Δu+up=0 in Ω, which vanish in a suitable trace sense on ∂Ω, but which are singular at prescribed isolated points if p is equal or slightly above . Similar constructions are carried out for solutions which are singular at any given embedded submanifold of ∂Ω of dimension k∈[0,N−2], if p equals or it is slightly above , and even on countable families of these objects, dense on a given closed set. The role of the exponent (first discovered by Brezis and Turner [H. Brezis, R. Turner, On a class of superlinear elliptic problems, Comm. Partial Differential Equations 2 (1977) 601-614]) for boundary regularity, parallels that of for interior singularities.     

On the existence of positive solutions for semilinear elliptic equations with indefinite nonlinearities

In this paper, we study the existence of positive solutions for the semilinear elliptic problem

     

Pressure representation and boundary regularity of the Navier-Stokes equations with slip boundary condition

We first represent the pressure in terms of the velocity in . Using this representation we prove that a solution to the Navier-Stokes equations is in under the critical assumption that , with r?3, while for r=3 the smallness is required. In [H.J. Choe, Boundary regularity of weak solutions of the Navier-Stokes equations, J. Differential Equations 149 (2) (1998) 211-247], a boundary L^∞ estimate for the solution is derived if the pressure on the boundary is bounded. In our work, we remove the boundedness assumption of the pressure. Here, our estimate is local. Indeed, employing Moser type iteration and the reverse Hölder inequality, we find an integral estimate for L^∞-norm of u.     

Semilinear elliptic variational inequalities with dependence on the gradient via Mountain Pass techniques

In this paper we consider a semilinear variational inequality with a gradient-dependent nonlinear term. Obviously the nature of this problem is non-variational. Nevertheless we study that problem associating a suitable semilinear variational inequality, variational in nature, with it, and performing an iterative technique used in De Figueiredo et al. (2004) [6] in order to treat semilinear elliptic equations when there is a gradient dependence on the nonlinearity. We prove the existence of a non-trivial non-negative weak solution u for our problem using essentially variational methods, a penalization technique and an iterative scheme. Via Lewy-Stampacchia’s estimates and regularity theory for elliptic equation we also show that u is differentiable and its gradient is α-H?lder continuous on for any α∈(0,1).     

Solutions for semilinear elliptic equations with critical exponents and Hardy potential

In this paper, we answer affirmatively an open problem (cf. Theorem 4′ in Ferrero and Gazzola (J. Differential Equations 177 (2001) 494): Let Ω∋0 be an open-bounded domain, Ω⊂RN(N?5) and assume that , then, for all λ>0 there exists a nontrivial solution with critical level in the range for the problem in Ω; u=0 on ∂Ω.     

Boundary blow-up rates of large solutions for elliptic equations with convection terms

By Karamata regular variation theory, a perturbation method and constructing comparison functions, we show the exact asymptotic behavior of large solutions to the semilinear elliptic equations with convection terms

     

Second-order semilinear elliptic inequalities in exterior domains

We study the problem of existence and nonexistence of positive solutions of the semilinear elliptic inequalities in divergence form with measurable coefficients in exterior domains in . For W(x)?|x|^−σ at infinity we compute the critical line on the plane (p,σ), which separates the domains of existence and nonexistence, and reveal the class of potentials V that preserves the critical line. Example are provided showing that the class of potentials is maximal possible, in certain sense. The case of (p,σ) on the critical line has also been studied.     

A priori bounds for semilinear equations and a new class of critical exponents for Lipschitz domains

A priori bounds for positive, very weak solutions of semilinear elliptic boundary value problems −Δu=f(x,u) on a bounded domain Ω⊂Rn with u=0 on ∂Ω are studied, where the nonlinearity 0?f(x,s) grows at most like sp. If Ω is a Lipschitz domain we exhibit two exponents p_* and p^*, which depend on the boundary behavior of the Green function and on the smallest interior opening angle of ∂Ω. We prove that for 1<p<p_* all positive very weak solutions are a priori bounded in L^∞. For p>p^* we construct a nonlinearity f(x,s)=a(x)sp together with a positive very weak solution which does not belong to L^∞. Finally we exhibit a class of domains for which p_*=p^*. For such domains we have found a true critical exponent for very weak solutions. In the case of smooth domains is an exponent which is well known from classical work of Brezis, Turner [H. Brezis, R.E.L. Turner, On a class of superlinear elliptic problems, Comm. Partial Differential Equations 2 (1977) 601-614] and from recent work of Quittner, Souplet [P. Quittner, Ph. Souplet, A priori estimates and existence for elliptic systems via bootstrap in weighted Lebesgue spaces, Arch. Ration. Mech. Anal. 174 (2004) 49-81].     

Upper and lower bounds for higher order eigenvalues of some semilinear elliptic equations

We prove upper and lower bounds on the eigenvalues (as the norm of the eigenfunction tends to zero) in bifurcation problems for a class of semilinear elliptic equations in bounded domains of R^N. It is shown that these bounds are computable in terms of the eigenvalues of the associated linear equation.     

The period function of the generalized Lotka-Volterra centers

The present paper deals with the period function of the quadratic centers. In the literature different terminologies are used to classify these centers, but essentially there are four families: Hamiltonian, reversible , codimension four Q₄ and generalized Lotka-Volterra systems . Chicone [C. Chicone, Review in MathSciNet, Ref. 94h:58072] conjectured that the reversible centers have at most two critical periods, and that the centers of the three other families have a monotonic period function. With regard to the second part of this conjecture, only the monotonicity of the Hamiltonian and Q₄ families [W.A. Coppel, L. Gavrilov, The period function of a Hamiltonian quadratic system, Differential Integral Equations 6 (1993) 1357-1365; Y. Zhao, The monotonicity of period function for codimension four quadratic system Q₄, J. Differential Equations 185 (2002) 370-387] has been proved. Concerning the family, no substantial progress has been made since the middle 80s, when several authors showed independently the monotonicity of the classical Lotka-Volterra centers [F. Rothe, The periods of the Volterra-Lokta system, J. Reine Angew. Math. 355 (1985) 129-138; R. Schaaf, Global behaviour of solution branches for some Neumann problems depending on one or several parameters, J. Reine Angew. Math. 346 (1984) 1-31; J. Waldvogel, The period in the Lotka-Volterra system is monotonic, J. Math. Anal. Appl. 114 (1986) 178-184]. By means of the first period constant one can easily conclude that the period function of the centers in the family is monotone increasing near the inner boundary of its period annulus (i.e., the center itself). Thus, according to Chicone's conjecture, it should be also monotone increasing near the outer boundary, which in the Poincaré disc is a polycycle. In this paper we show that this is true. In addition we prove that, except for a zero measure subset of the parameter plane, there is no bifurcation of critical periods from the outer boundary. Finally we show that the period function is globally (i.e., in the whole period annulus) monotone increasing in two other cases different from the classical one.