Asymptotic Behavior Of Nonnegative Solutions Research Articles

Bôcher-type results for the fourth and higher order equations on singular manifolds with conical metrics

We obtain the Bocher-type theorems and present the sharp characterization of the asymptotic behavior at the isolated singularities of solutions of some fourth and higher order equations on singular manifolds with conical metrics. It is seen that the equations on singular manifolds with conical metrics are equivalent to weighted elliptic equations in \begin{document}$ B \backslash \{0\} $\end{document} , where \begin{document}$ B \subset \mathbb{R}^N $\end{document} is the unit ball. The weights can be singular at \begin{document}$ x = 0 $\end{document} . We present the sharp asymptotic behavior of nonnegative solutions of the weighted elliptic equations near \begin{document}$ x = 0 $\end{document} and the Liouville-type results for the degenerate elliptic equations in \begin{document}$ \mathbb{R}^N \backslash \{0\} $\end{document} .

Critical quasilinear elliptic problems using concave–convex nonlinearities

It is established existence, multiplicity, and asymptotic behavior of nonnegative solutions for a quasilinear elliptic problem driven by the $$\Phi $$ -Laplacian operator. One of these solutions is obtained as ground-state solution by applying the well-known Nehari method. The nonlinear term is a concave–convex function which presents a critical behavior at infinity. The concentration-compactness principle is used in order to recover the compactness required in variational methods.

Fractional Schrödinger–Poisson–Kirchhoff type systems involving critical nonlinearities

The paper is concerned with existence, multiplicity and asymptotic behavior of nonnegative solutions for a fractional Schrödinger–Poisson–Kirchhoff type system. As a consequence, the results can be applied to the special case (a+b‖u‖2)[(−Δ)su+V(x)u]+ϕk(x)|u|p−2u=λh(x)|u|q−2u+|u|2s∗−2uinR3,(−Δ)tϕ=k(x)|u|pinR3,‖u‖=∫R3∫R3|u(x)−u(y)|2|x−y|3+2sdxdy+∫R3V(x)|u|2dx1∕2,where a,b≥0 are two numbers, with a+b>0, 1<p<2s,t∗=3+2t3−2s, λ>0 is a parameter, s,t∈(0,1), (−Δ)s is the fractional Laplacian, k∈L63+2t−p(3−2s)(R3) may change sign, V:R3→[0,∞) is a potential function, 2s∗=6∕(3−2s) is the critical Sobolev exponent, 1<q<2s∗ and h∈L2s∗2s∗−q(R3). First, when θ<p<2s∗∕2, 2p≤q<2s∗ and λ is large enough, existence of nonnegative solutions is obtained by the mountain pass theorem. Moreover, we obtain that limλ→∞‖uλ‖=0. Then, via the Ekeland variational principle, existence of nonnegative solutions is investigated when θ<p<2s∗∕2, 1<q<2 and λ is small enough, and we obtain that limλ→0‖uλ‖=0. Finally, we consider the system with double critical exponents, that is, p=2s,t∗, and obtain two nontrivial and nonnegative solutions in which one is least energy solution and another is mountain pass solution. The paper covers a novel feature of Kirchhoff problems, which is the fact that the parameter a can be zero. Hence the results of the paper are new even for the standard stationary Schrödinger–Poisson–Kirchhoff system.

Asymptotic expansion of solutions to the Cauchy problem for doubly degenerate parabolic equations with measurable coefficients

We study the asymptotic behaviour of nonnegative solutions of the Cauchy problem for doubly degenerate parabolic equations with variable coefficients. When the initial datum has a finite mass, the asymptotic expansion of the solution for a large time, uniformly in whole space, is established.

Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity

This paper deals with the existence and the asymptotic behavior of non-negative solutions for a class of stationary Kirchhoff problems driven by a fractional integro-differential operator LK and involving a critical nonlinearity. In particular, we consider the problem −M(||u||2)LKu=λf(x,u)+|u|2s∗−2uin Ω,u=0in Rn∖Ω, where Ω⊂Rn is a bounded domain, 2s∗ is the critical exponent of the fractional Sobolev space Hs(Rn), the function f is a subcritical term and λ is a positive parameter. The main feature, as well as the main difficulty, of the analysis is the fact that the Kirchhoff function M could be zero at zero, that is the problem is degenerate. The adopted techniques are variational and the main theorems extend in several directions previous results recently appeared in the literature.

Multiplicity and asymptotic behavior of nonnegative solutions for elliptic problems involving nonlinearities indefinite in sign

Let p,r,s,λ,μ be positive numbers with 1<r<s<p. The multiplicity of nonnegative solutions for the problem −Δpu=λus−1−μur−1 in Ω, u∣∂Ω=0, is investigated. We prove that there exist two positive constants Λ2,rs,Λrs with Λ2,rs<Λrs such that the above problem has at least two solutions, at least one solution or no solutions according to whether μλ−p−rp−s≤Λ2,rs, μλ−p−rp−s∈]Λ2,rs,Λrs] or μλ−p−rp−s>Λrs. We also study the asymptotic behavior of the solutions as μ→0+.

Very slow convergence to zero for a supercritical semilinear parabolic equation

We study the asymptotic behavior of nonnegative solutions to the Cauchy problem for a semilinear parabolic equation with a supercritical nonlinearity. It is known that there are initial data such that the corresponding solution decays to zero with an algebraic rate. Furthermore, any algebraic rate which is slower than the self-similar rate occurs as decay rate for some solution. In this paper we prove that the convergence to zero can take place with an "arbitrarily" slow rate, if the initial data are chosen properly.

Lubrication Approximation for Thin Viscous Films: Asymptotic Behavior of Nonnegative Solutions

We use standard regularized equations and adapted entropy functionals to prove exponential asymptotic decay in the H 1 norm for nonnegative weak solutions of fourth-order nonlinear degenerate parabolic equations of lubrication approximation for thin viscous film type. The weak solutions considered arise as limits of solutions for the regularized problems. Relaxed problems, with second-order nonlinear terms of porous media type are also successfully treated by the same means. The problems investigated here are one-dimensional in space, with power-law nonlinearities. Our approach is direct and natural, as it is adapted to deal with the more complex nonlinear terms occurring in the regularized, approximating problems.

Blow-up with logarithmic nonlinearities

We study the asymptotic behaviour of nonnegative solutions of the nonlinear diffusion equation in the half-line with a nonlinear boundary condition, { u t = u x x − λ ( u + 1 ) log p ( u + 1 ) , ( x , t ) ∈ R + × ( 0 , T ) , − u x ( 0 , t ) = ( u + 1 ) log q ( u + 1 ) ( 0 , t ) , t ∈ ( 0 , T ) , u ( x , 0 ) = u 0 ( x ) , x ∈ R + , with p , q , λ > 0 . We describe in terms of p, q and λ when the solution is global in time and when it blows up in finite time. For blow-up solutions we find the blow-up rate and the blow-up set and we describe the asymptotic behaviour close to the blow-up time, showing that a phenomenon of asymptotic simplification takes place. We finally study the appearance of extinction in finite time.

“A priori” estimates, uniqueness and existence of positive solutions of Yamabe type equations on complete manifolds

We study the asymptotic behaviour of non-negative solutions of Yamabe type equations on a complete Riemannian manifold. Then we provide a comparison result, based on a form of the weak maximum principle at infinity, which together with the “a priori” estimates previously obtained, yields uniqueness under very general Ricci assumptions. The paper ends with an existence result and an application to the non-compact Yamabe problem.

Nonlinear diffusions and optimal constants in Sobolev type inequalities: asymptotic behaviour of equations involving the [formula omitted]-Laplacian

We study the asymptotic behaviour of nonnegative solutions to: u t = Δ p u m using an entropy estimate based on a sub-family of the Gagliardo–Nirenberg inequalities – or, in the limit case m=( p−1) −1, on a logarithmic Sobolev inequality in W 1, p – for which optimal functions are known. To cite this article: M. Del Pino, J. Dolbeault, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 365–370.

Nonexistence results of positive solutions for semilinear elliptic equations in Rn

We consider the global properties of nonnegative solutions of the semilinear elliptic equations in the entire space. By employing Pohozaev identity in the entire space and the results concerning the asymptotic behavior of nonnegative solutions, we establish some theorems of Liouville type.

Large time behaviour of the doubly nonlinear porous medium equation

The main goal of this paper is to study the asymptotic behaviour of nonnegative solutions of the general porous medium equation in the slow diffusion case as well as in the limit case when diffusion loses its slow character. We show that these solutions approach the unique solution of the associated steady state problem when t goes to infinity.

Fundamental Solutions and Asymptotic Behaviour for the $p$-Laplacian Equation

We establish the uniqueness of fundamental solutions to the p -Laplacian equation \mathrm {(PLE)} \; u_t = \mathrm {div} (|Du|^{p-2}Du), \; p &gt; 2, defined for x \in \mathbb R^N , 0 &lt; t &lt; T . We derive from this result the asymptotic behaviour of nonnegative solutions with finite mass, i.e. such that u(\cdotp, t) \in L^1(\mathbb R^N) . Our methods also apply to the porous medium equation \mathrm {(PME)} \; u_t = \Delta (u^m), \; m &gt; 1, giving new and simpler proofs of known results. We finally introduce yet another method of proving asymptotic results based on the idea of asymptotic radial symmetry. This method can be useful in dealing with more general equations.

On a result of K. Masuda concerning reaction-diffusion equations

We give a simplified proof of a recent result due to K. Masuda concerning the global existence and asymptotic behavior of non-negative solutions to some reaction-diffusion systems. This new method also provides an analogous result under weaker growth assumptions on the nonlinear terms. Introduction. Let Ω be an open, bounded domain of class C1 in Rn, with boundary Γ — dΩ. Let dlf d2 be two positive constants with dγ Φ d2 and a^x), a2{x) two nonnegative functions of class C\Γ) with a^^l, a2 tί 1. Let φeC\R+) be a nonnegative function. We consider the reaction-diffusion system (du/dt — dλAu + uφ{v) = 0 on R+xΩ \dv/dt — d2Δv — uφ{v) = 0 on R+xΩ with the homogeneous boundary conditions {aβu/dn + (1 - aλ)u = 0 on R+xΓ (2)

On the Asymptotic Behaviour of Nonnegative Solutions of a Certain Integral Inequality

On the Asymptotic Behaviour of Nonnegative Solutions of a Certain Integral Inequality

On the asymptotic behaviour of nonnegative solutions of a certain integral inequality

The asymptotic behaviour of nonnegative solutions of a certain integral inequality is discussed, in the framework of a probabilistic-potential theoretic boundary theory.