On blow-up at space infinity for semilinear heat equations

A nonnegative blowing up solution of the semilinear heat equation ut=Δu+up with p>1 is considered when initial data u₀ satisfies

     

有限初始能量的相对论欧拉方程组光滑解的爆破

在初始资料的某些限制下证明有限初始能量的相对论欧拉方程组柯西问题光滑解的爆破.该文的爆破条件不需要初始资料具有紧支集,部分补充了Pan和Smoller的经典爆破结果(2006).     

Blowup of classical solutions to the pressureless Euler/isentropic Navier‐Stokes equations

In this paper, we will show the blowup of classical solutions to the Cauchy problem for the pressureless Euler/isentropic Navier‐Stokes equations in arbitrary dimensions under some restrictions on the initial data. Compared with the degenerate viscosities appeared in the recent work, we consider the constant viscosities, but we can remove the condition that the adiabatic exponent has a upper bound, which was a key constraint in the proof of the blow‐up result is based on the construction of some new differential inequalities.     

Blowup of solutions for the initial boundary value problem of the 3‐dimensional compressible damped Euler equations

The blowup phenomenon of solutions is investigated for the initial boundary value problem of the 3‐dimensional compressible damped Euler equations. It is shown that if the initial functional associated with a general test function is large enough, the solutions of the damped Euler equations will blow up on finite time. Hence, a class of blowup conditions is established. Moreover, the blowup time could be estimated.     

Multiple solutions for a degenerate elliptic equation involving sublinear terms at infinity

Some multiplicity results are presented for the eigenvalue problem

(Pλ,μ)     

Multiple solutions for elliptic equations at resonance

We study an asymptotically linear elliptic equation at resonance, with an odd nonlinearity. By a penalization technique and suitable min-max theorems (which give Morse index estimates), we prove the existence of pairs of non trivial solutions, where N is, roughly speaking, the difference between the Morse indexes at zero and at infinity. Received December 1999     

On blow-up solutions for systems of semilinear heat equations with quadratic nonlinearities

This article deals with blow-up solutions of the Cauchy–Dirichlet problem for system of semilinear heat equations with quadratic non-linearities. Sufficient conditions for the existence of blow-up solutions are established. Sets of initial values for these solutions as well as upper bounds for corresponding blow-up time are determined. Furthermore, an application to the Lotka-Volterra system with diffusion is also discussed. The result of this article may be considered as a continuation and a generalization of the results obtained in (Baris, J., Baris, P. and Ruchlewicz, B., 2006, On blow-up solutions of nonautonomous quadratic differential systems. Differential Equations, 42, 320–326; Baris, J., Baris, P. and Wawiórko, E., 2006, Asymptotic behaviour of solutions of Lotka-Volterra systems. Nonlinear Analysis: Real World Applications, 7, 610–618; Baris, J., Baris, P. and Ruchlewicz, B., 2006, On blow-up solutions of quadratic systems of differential equations. Sovremennaya Matematika. Fundamentalnye Napravleniya, 15, 29–35 (in Russian); Baris, J. and Wawiórko, E., On blow-up solutions of polynomial Kolmogorov systems. Nonlinear Analysis: Real World Applications, to appear).     

Boundedness of semilinear Duffing equations at resonance in a critical situation

In this paper, we propose a sufficient and necessary condition for the boundedness of all the solutions for the equation $\ddot{x}+n^{2}x+g(x)=p(t)$ with the critical situation that $|{\int }_0^{2\pi }p(t)e^{?int}dt|=2|g(+\infty )?g(?\infty )|$ on g and p, where $n\in {\mathbb{N}}^{+}$, $p(t)$ is periodic and $g(x)$ is bounded.     

Global existence and blowup of solutions for nonlinear coupled wave equations with viscoelastic terms

In this paper, we study a system of nonlinear coupled wave equations with damping, source, and nonlinear strain terms. We obtain several results concerning local existence, global existence, and finite time blow‐up property with positive initial energy by using Galerkin method and energy method, respectively. Copyright © 2015 John Wiley & Sons, Ltd.     

Infinitely many nodal solutions with a prescribed number of nodes for the Kirchhoff type equations

In this paper, we investigate the existence of multiple radial sign-changing solutions with the nodal characterization for a class of Kirchhoff type problems$\{\begin{array}{cc}\hfill & ?(a+b|\mathrm{?}u{|}_{L^2}^2)\mathrm{\Delta }u+V(|x|)u=K(|x|)f(u)\text{in}{\mathbb{R}}^N,\hfill \\ \hfill & u\in H^1({\mathbb{R}}^N),\hfill \end{array}$ where $N=1,2,3,a,b>0$, $V,K$ are radial and bounded away from below by positive numbers. Under some weak assumptions on $f\in C^0(\mathbb{R};\mathbb{R})$, by taking advantage of the Gersgorin disc's theorem and Miranda theorem, we develop some new analytic techniques and prove that this problem admits infinitely many nodal solutions $\{U_k^b\}$ having a prescribed number of nodes k, whose energy is strictly increasing in k. Moreover, the asymptotic behaviors of $U_k^b$ as $b\to 0_{+}$ are established. These results improve and generalize the previous results in the literature.     

Blowup behavior of solutions for a semilinear heat equation with supercritical nonlinearity

This paper is concerned with blowup phenomena of solutions for the Cauchy and the Cauchy-Dirichlet problem of

(P)     

Dominant and recessive solutions at infinity and genera of conjoined bases for discrete symplectic systems

In this paper we introduce the theory of dominant solutions at infinity for nonoscillatory discrete symplectic systems without any controllability assumption. Such solutions represent an opposite concept to recessive solutions at infinity, which were recently developed for such systems by the authors. Our main results include: (i) the existence of dominant solutions at infinity for all ranks in a given range depending on the order of abnormality of the system, (ii) construction of dominant solutions at infinity with eventually the same image, (iii) classification of dominant and recessive solutions at infinity with eventually the same image, (iv) limit characterization of recessive solutions at infinity in terms of dominant solutions at infinity and vice versa, and (v) Reid’s construction of the minimal recessive solution at infinity. These results are based on a new theory of genera of conjoined bases for symplectic systems developed for this purpose in this paper.     

Multiplicity of nontrivial solutions for system of nonhomogenous Kirchhoff‐type equations in RN

In this paper, we are interested in looking for multiple solutions for the following system of nonhomogenous Kirchhoff‐type equations: (1.1) where constants a,c > 0;b,d,λ≥0, N = 1,2 or 3, f,g∈L²(R^N) and f,g?0, F∈C¹(R^N×R²,R), , V∈C(R^N,R) satisfy some appropriate conditions. Under more relaxed assumptions on the nonlinear term F, the existence of one negative energy solution and one positive energy solution for the nonhomogenous system 2.1 is obtained by Ekeland's variational principle and Mountain Pass Theorem, respectively. Copyright © 2014 John Wiley & Sons, Ltd.     

Convergent solutions of linear functional difference equations in phase space

By using weighted summable dichotomies and Schauder's fixed point theorem, we prove the existence of convergent solutions of linear functional difference equations. We apply our result to Volterra difference equations with infinite delay.     

Blowup rate of solutions for a semilinear heat equation with the Neumann boundary condition

Let p>1 and Ω be a smoothly bounded domain in . This paper is concerned with a Cauchy-Neumann problem

     

Positive periodic solutions for nonlinear repulsive singular difference equations

The existence and multiplicity of positive solutions are established to the periodic boundary value problems for repulsive singular nonlinear difference equations. The proof relies on a nonlinear alternative of Leray–Schauder type and on Krasnoselskii fixed point theorem on compression and expansion of cones.