Global existence and blowing-up of solutions to a class of coupled reaction–convection–diffusion systems

This paper concerns the global existence and blowing-up of solutions to the homogeneous Neumann problem of a coupled reaction–convection–diffusion system. The critical Fujita curve is determined and blowing-up theorem of Fujita type is established. An interesting phenomenon is that the critical Fujita curve even could be the infinite due to the convection.     

On Bounded Solutions of the Emden-Fowler Equation in a Semi-cylinder

Bounded solutions of the Emden-Fowler equation in a semi-cylinder are considered. For small solutions the asymptotic representations at infinity are derived. It is shown that there are large solutions whose behavior at infinity is different. These solutions are constructed when some inequalities between the dimension of the cylinder and the homogeneity of the nonlinear term are fulfilled. If these inequalities are not satisfied then it is proved, for the Dirichlet problem, that all bounded solutions tend to zero and have the same asymptotics as small solutions.     

The asymptotic behavior of solutions of a certain nonlinear Volterra equation

We study the asymptotic behavior of bounded and unbounded solutions to the Volterra-Hammerstein equation. We obtain conditions for the admissibility of a pair of spaces consisting of the sum of a quasipolynomial and the Taylor expansion at infinity.     

Lazer-McKenna conjecture: The critical case

We consider an elliptic problem of Ambrosetti-Prodi type involving critical Sobolev exponent on a bounded smooth domain of dimension six or higher. By constructing solutions with many sharp peaks near the boundary of the domain, but not on the boundary, we prove that the number of solutions for this problem is unbounded as the parameter tends to infinity, thereby proving the Lazer-McKenna conjecture in the critical case.     

On the exponential decay of the critical generalized Korteweg-de Vries equation with localized damping

This paper is concerned with the asymptotic behavior of solutions of the critical generalized Korteweg-de Vries equation in a bounded interval with a localized damping term. Combining multiplier techniques and compactness arguments it is shown that the problem of exponential decay of the energy is reduced to prove the unique continuation property of weak solutions. A locally uniform stabilization result is derived.

     

Self-similar solutions of a generalized Burgers equation with nonlinear damping

The nonlinear ordinary differential equation resulting from the self-similar reduction of a generalized Burgers equation with nonlinear damping is studied in some detail. Assuming certain asymptotic conditions at plus infinity or minus infinity, we find a wide variety of solutions—(positive) single hump, monotonic (bounded or unbounded) or solutions with a finite zero. The existence or non-existence of positive bounded solutions with exponential decay to zero at infinity for specific parameter ranges is proved. The analysis relies mainly on the shooting argument.     

Qualitative analysis of families of bounded solutions of the nonlinear three-dimensional schrödinger equation

It is well known that the nonlinear three-dimensional Schrödinger equation with a power-law nonlinearity can be reduced to a set of ordinary differential equations. In the present paper we analyze the problem concerning existence and asymptotic behavior of solutions of these equations bounded on a semiaxis. Based on our approach, we describe families of solutions of the Schrödinger equation possessing various properties: quasi-periodic, spherically-symmetric, decreasing at infinity with respect to the spatial variables, and unbounded growth with time.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 10, pp. 1344–1349, October, 1990.     

The asymptotic behavior of the strong solutions for a non-autonomous non-local PDE model with delay

In this paper, we study the asymptotic behavior of the strong solutions of a non-autonomous non-local PDE model with time delay. We present the existence and structure of the uniform attractor by constructing the skew product flow of the family of processes generated by the strong solutions. In order to obtain the existence of the uniform attractor, we prove the family of processes satisfies uniform condition (C) by using some special technique of phase space decomposition. Additionally, it is shown that all the bounded complete trajectories are globally asymptotic stable under some assumptions. As the application of our result, we obtain a globally asymptotic stable nontrivial strong periodic solution of a non-local PDE model.     

The exterior Dirichlet problem for a quasilinear elliptic boundary value problem suggested by plane shear flow

We study the existence of a classical solution of the exterior Dirichlet problem for a class of quasilinear elliptic boundary value problems that are suggested by plane shear flow. In this connection only bounded solutions which tend to zero at infinity are of interest. A priori bounds on solutions and constructive existence proofs are given. Finally, we prove the existence of a unique bounded solution of the shear flow and we show, under certain hypotheses about the asymptotic behavior of the nonlinearity, that this solution tends to zero at infinity. As an example, we consider the case of the parabolic shear flow.     

Solvability of some classes of nonlinear integro-differential equations with noncompact operator

For a class of nonlinear integrodifferential equations with a noncompact Urysohn-type operator we prove the existence of nonnegative bounded solutions. We study the asymptotic behavior of solutions at infinity. We give some examples that are of practical interest.     

On the minimal solution for some variational inequalities

Applying an asymptotic method, the existence of the minimal solution to some variational elliptic inequalities defined on bounded or unbounded domains is established. The minimal solution is obtained as limit of solutions to some classical variational inequalities defined on domains becoming unbounded when some parameter tends to infinity. The considered quasilinear operators are only monotone (not strictly) and noncoercive. Some related comparison principles are also investigated.     

Large time behavior of solutions of Hamilton–Jacobi equations with periodic boundary data

We study the large time behavior of viscosity solutions of Hamilton–Jacobi equations with periodic boundary data on bounded domains. We establish a result on convergence of viscosity solutions to state constraint asymptotic solutions or periodic asymptotic solutions depending on the sign of critical value as time goes to infinity.     

Fujita type critical exponent for a free boundary problem with spatial–temporal source

In this paper, we investigate a nonlinear free boundary problem incorporating with nontrivial spatial and exponential temporal weighted source. To portray the asymptotic behavior of the solution, we first derive some sufficient conditions for finite time blowup. Furthermore, the global vanishing solution is also obtained for a class of small initial data. Finally, a sharp threshold trichotomy result is provided in terms of the size of the initial data to distinguish the blowup solution, the global vanishing solution, and the global transition solution. In particular, our results show that such a problem always possesses a Fujita type critical exponent whenever the spatial source is just equivalent to a trivial constant, or is an extreme one, such as “very negative” one in the sense of measure or integral.     

Quenching of solutions to a class of semilinear parabolic equations with boundary degeneracy

This paper concerns the quenching phenomenon of solutions to a class of semilinear parabolic equations with boundary degeneracy. In the case that the degeneracy is not strong, it is shown that there exists a critical length, which is positive, such that the solution exists globally in time if the length of the spatial interval is less than it, while quenches in a finite time if the length of the spatial interval is greater than it. Whereas in the case that the degeneracy is strong enough, the solution must be quenching in a finite time no matter how long the spatial interval is. Furthermore, for each quenching solution, the set of quenching points is determined and it is proved that its derivative with respect to the time must blow up at the quenching time.     

A note on semilinear heat equation in hyperbolic space

Bandle et al. [1] obtained a quite interesting result about a semilinear heat equation that the Fujita exponent relative to the whole hyperbolic space is just the same as that relative to bounded domain in Euclidean space, and, in addition, the properties of solutions are different in the critical exponent case. Our purpose is to answer an open problem proposed by Bandle et al. for the critical exponent case, and it, together with the one obtained by them, shows that the critical exponent case does belong to the non-blow-up case, which is completely different from the case in Euclidean space.     

Nontrivial solutions of Kirchhoff type problems

In this work, we study Kirchhoff type problems on a bounded domain. We consider the cases where the nonlinearity is superlinear near zero but asymptotically 4-linear at infinity, and the nonlinearity is asymptotically linear near zero but 4-superlinear at infinity. By computing the relevant critical groups, we obtain nontrivial solutions via Morse theory.     

On the Number of Unbounded Solution Branches in a Neighborhood of an Asymptotic Bifurcation Point

We suggest a method for studying asymptotically linear vector fields with a parameter. The method permits one to prove theorems on asymptotic bifurcation points (bifurcation points at infinity) for the case of double degeneration of the principal linear part. We single out a class of fields that have more than two unbounded branches of singular points in a neighborhood of a bifurcation point. Some applications of the general theorems to bifurcations of periodic solutions and subharmonics as well as to the two-point boundary value problem are given.     

Solutions Almost Periodic at Infinity to Differential Equations With Unbounded Operator Coefficients

The new class of functions almost periodic at infinity is defined using the subspace of functions with integrals decreasing at infinity. We obtain spectral criteria for almost periodicity at infinity of bounded solutions to differential equations with unbounded operator coefficients. For the new class of asymptotically finite operator semigroups we prove the almost periodicity at infinity of their orbits.     

Semidiscretization in Space of Nonlinear Degenerate Parabolic Equations with Blow-up of the Solutions

Semidiscretization in space of nonlinear degenerate parabolic equations of nondivergent form is presented, under zero Dirichlet boundary condition. It is shown that semidiscrete solutions blow up in finite time. In particular, the asymptotic behavior of blowing-up solutions, is discussed precisely.     

Asymptotic Analysis of a Continuous-Discrete Control System with a Singularity in the Coefficients

The article is concerned with the asymptotic analysis of solutions of a continuous-discrete endpoint control system as the operating interval tends to infinity. The author obtains criteria of stable and unstable behavior of the system in the basic and critical cases.