On critical Fujita exponents for the porous medium equation with a nonlinear boundary condition

We establish the critical Fujita exponents for the solution of the porous medium equation u_t=Δu^m, x∈R₊^N, t>0, subject to the nonlinear boundary condition −∂u^m/∂x₁=u^p, x₁=0, t>0, in multi-dimension.     

Positive solutions for some Schrödinger equations having partially periodic potentials

This paper is concerned with the problem of finding positive solutions of the equation −Δu+(a_∞+a(x))u=|u|^q−2u, where q is subcritical, Ω is either RN or an unbounded domain which is periodic in the first p coordinates and whose complement is contained in a cylinder , a_∞>0, a∈C(RN,R) is periodic in the first p coordinates, infx∈RN(a_∞+a(x))>0 and a(x^′,x^″)→0 as |x^″|→∞ uniformly in x^′. The cases a?0 and a?0 are considered and it is shown that, under appropriate assumptions on a, the problem has one solution in the first case and p+1 solutions in the second case when p?N−2.     

Attractors for reaction-diffusion equations on thin domains whose linear part is non-self-adjoint

For a bounded smooth domain Ω⊂R^N_x+N_y let Ω_?, 0<?, be a family of domains squeezed in y∈R^N_y direction. On Ω_? we consider a reaction-diffusion equation with nonsymmetrical linear part. We show that under natural conditions on the nonlinearity the generated semi-flows have global attractors which in a certain sense have limits, as ?↓0.     

Quantum Dynamical coBoundary Equation for finite dimensional simple Lie algebras

For a finite dimensional simple Lie algebra g, the standard universal solution R(x)∈Uq(g)^⊗2 of the Quantum Dynamical Yang-Baxter Equation quantizes the standard trigonometric solution of the Classical Dynamical Yang-Baxter Equation. It can be built from the standard R-matrix and from the solution F(x)∈Uq(g)^⊗2 of the Quantum Dynamical coCycle Equation as . F(x) can be computed explicitly as an infinite product through the use of an auxiliary linear equation, the ABRR equation.Inspired by explicit results in the fundamental representation, it has been conjectured that, in the case where g=sl(n+1)(n?1) only, there could exist an element M(x)∈Uq(sl(n+1)) such that the dynamical gauge transform RJ of R(x) by M(x),

RJ=M₁⁻¹(x)M₂(xqh₁⁾⁻¹R(x)M₁(xqh₂)M₂(x),     

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.     

Universal self-similarity of porous media equation with absorption: the critical exponent case

In this paper we study the large time behavior of non-negative solutions to the Cauchy problem of u_t=Δu^m−u^q in R^N×(0,∞), where m>1 and q=q_c≡m+2/N is a critical exponent. For non-negative initial value u(x,0)=u₀(x)∈L¹(R^N), we show that the solution converges, if u₀(x)(1+|x|)^k is bounded for some k>N, to a unique fundamental solution of u_t=Δu^m, independent of the initial value, with additional logarithmic anomalous decay exponent in time as t→∞.     

Boundary behaviour of the unique solution to a singular Dirichlet problem with a convection term

By Karamata regular variation theory and constructing comparison functions, we derive that the boundary behaviour of the unique solution to a singular Dirichlet problem −Δu=b(x)g(u)+λq|∇u|, u>0, x∈Ω, u_|∂Ω=0, which is independent of λq|∇uλ|, where Ω is a bounded domain with smooth boundary in RN, λ∈R, q∈(0,2], lims→⁰⁺g(s)=+∞, and b is non-negative on Ω, which may be vanishing on the boundary.     

The continuous dependence on parameters of solutions for a class of elliptic problems on exterior domains

The goal of this paper is to discuss the continuous dependence of solutions on functional parameters for the following semilinear elliptic partial differential equation: , for x∈Ω_r0?{x∈Rⁿ,n≥3,‖x‖>r₀} and v∈V, where V stands for some functional space. Our approach covers the case when f may change sign and admits general growth. As an additional result, the characterization of the radius r₀ for which our problem possesses at least one positive evanescent solution in the exterior domain Ω_r0 is described and numerically illustrated. Our approach relies on the subsolution and supersolution method and on a lemma due to Noussair and Swanson.     

类渐近线性p&q-Laplace方程弱解的全局衰减性

该文研究椭圆型方程 {Δ_pu+m|u|^p-2u-Δ_qu+n|u|^q-2u=g(x, u), x∈R^N, u∈ W^{1, p}(R^N)∩W^{1, q}(R^N) 弱解在全空间R^N上的衰减性, 其中m, n ≥ 0, N≥3, 1 < q < p < N, g(x, u)关于u满足类渐近线性. 证明了该方程的 弱解在无穷远处关于|x|呈指数衰减性.     

Properties of the density for a three-dimensional stochastic wave equation

We consider a stochastic wave equation in space dimension three driven by a noise white in time and with an absolutely continuous correlation measure given by the product of a smooth function and a Riesz kernel. Let pt,x(y) be the density of the law of the solution u(t,x) of such an equation at points (t,x)∈]0,T]×R³. We prove that the mapping (t,x)?pt,x(y) owns the same regularity as the sample paths of the process {u(t,x),(t,x)∈]0,T]×R³} established in [R.C. Dalang, M. Sanz-Solé, Hölder-Sobolev regularity of the solution to the stochastic wave equation in dimension three, Mem. Amer. Math. Soc., in press]. The proof relies on Malliavin calculus and more explicitly, the integration by parts formula of [S. Watanabe, Lectures on Stochastic Differential Equations and Malliavin Calculus, Tata Inst. Fund. Res./Springer-Verlag, Bombay, 1984] and estimates derived from it.     

On solvability of two-dimensional Lp-Minkowski problem

Given p≠0 and a positive continuous function g, with g(x+T)=g(x), for some 0<T<1 and all real x, it is shown that for suitable choice of a constant C>0 the functional has a minimizer in the class of positive functions u∈C¹(R) for which u(x+T)=u(x) for all x∈R. This minimizer is used to prove the existence of a positive periodic solution y∈C²(R) of two-dimensional L_p-Minkowski problem y^1−p(x)(y″(x)+y(x))=g(x), where p∉{0,2}.     

Which generalized Randi? indices are suitable measures of molecular branching?

Molecular branching is a very important notion, because it influences many physicochemical properties of chemical compounds. However, there is no consensus on how to measure branching. Nevertheless two requirements seem to be obvious: star is the most branched graph and path is the least branched graph. Every measure of branching should have these two graphs as extremal graphs. In this paper we restrict our attention to chemical trees (i.e. simple connected graphs with maximal degree at most 4), hence we have only one requirement that the path be an extremal graph. Here, we show that the generalized Randi? index R_p(G)=∑_uv∈E(G)(d_ud_v)^p is a suitable measure for branching if and only if p∈[λ,0)∪(0,λ^′) where λ is the solution of the equation in the interval (−0.793,−0.792) and λ^′ is the positive solution of the equation 3⋅3^x−2⋅2^x−4^x=0. These results include the solution of the problem proposed by Clark and Gutman.     

Global well-posedness for the viscous Camassa-Holm equation

We consider the initial value problem (IVP) of the Camassa-Holm equation with viscosity. We established global solution for the IVP with u₀∈L²(R). This result improves the previous results.     

Regularity for a Schrödinger equation with singular potentials and application to bilinear optimal control

We study the Schrödinger equation i∂_tu+Δu+V₀u+V₁u=0 on R³×(0,T), where V₀(x,t)=|x-a(t)|^-1, with a∈W^2,1(0,T;R³), is a coulombian potential, singular at finite distance, and V₁ is an electric potential, possibly unbounded. The initial condition u₀∈H²(R³) is such that . The potential V₁ is also real valued and may depend on space and time variables. We prove that if V₁ is regular enough and at most quadratic at infinity, this problem is well-posed and the regularity of the initial data is conserved for the solution. We also give an application to the bilinear optimal control of the solution through the electric potential.     

Scalar conservation laws with general boundary condition and continuous flux function

We introduce a notion of entropy solution for a scalar conservation law on a bounded domain with nonhomogeneous boundary condition: ut+divΦ(u)=f on Q=(0,T)×Ω, u(0,⋅)=u₀ on Ω and “u=a on some part of the boundary (0,T)×∂Ω.” Existence and uniqueness of the entropy solution is established for any Φ∈C(R;RN), u₀∈L^∞(Ω), f∈L^∞(Q), a∈L^∞((0,T)×∂Ω). In the L¹-setting, a corresponding result is proved for the more general notion of renormalised entropy solution.     

About the p-adic Yosida equation inside a disk

Let K be a complete ultrametric algebraically closed field and let ?(d(0, R^?)) be the field of meromorphic functions inside the disk d(0,R⁻) = {x ∈ K ∣ ∣x∣ < R}. Let ?_b(d(0, R^?)) be the subfield of bounded meromorphic functions inside d(0,R⁻) and let ?_u(d(0, R^?)) = ?(d(0, R^?)) ? ?_b(d(0, R^?)) be the subset of unbounded meromorphic functions inside d(0,R⁻). Initially, we consider the Yosida Equation: , where m ∈ ?^* and F(X) is a rational function of degree d with coefficients in ?_b(d(0, R^?)). We show that, if d ≥ 2m + 1, this equation has no solution in ?_u(d(0, R^?)).Next, we examine solutions of the above equation when F(X) is apolynomial with constant coefficients and show that it has no unbounded analytic functions in d(0,R⁻). Further, we list the only cases when the equation may eventually admit solutions in ?_u(d(0, R^?)). Particularly, the elliptic equation may not.     

On ground state solutions for singular and semi-linear problems including super-linear terms at infinity

We establish a result concerning the existence of entire, positive, classical and bounded solutions which converge to zero at infinity for the semi-linear equation −Δu=λf(x,u),x∈R^N, where f:R^N×(0,∞)→[0,∞) is a suitable function and λ>0 is a real parameter. This result completes the principal theorem of A. Mohammed [A. Mohammed, Ground state solutions for singular semi-linear elliptic equations, Nonlinear Analysis (2008) doi:10.1016/j.na.2008.11.080] mainly because his result does not address the super-linear terms at infinity. Penalty arguments, lower-upper solutions and an approximation procedure will be used.     

Hardy-Sobolev-Maz'ya type equations in bounded domains

We study the regularity, Palais-Smale characterization and existence/nonexistence of solutions of the Hardy-Sobolev-Maz'ya equation in a bounded domain in RN where x∈RN is denoted as x=(y,z)∈Rk×RN−k and . We show different behaviors of PS sequences depending on t=0 or t>0.     

Existence and uniqueness of positive solution for nonhomogeneous sublinear elliptic equations

In this paper, we study the existence and the uniqueness of positive solution for the sublinear elliptic equation, −Δu+u=p|u|sgn(u)+f in RN, N?3, 0<p<1, f∈L²(RN), f>0 a.e. in RN. We show by applying a minimizing method on the Nehari manifold that this problem has a unique positive solution in H¹(RN)∩Lp+1(RN). We study its continuity in the perturbation parameter f at 0.