Blow‐up solutions for localized reaction‐diffusion equations with variable exponents

This paper deals with radial solutions to localized reaction‐diffusion equations with variable exponents, subject to homogeneous Dirichlet boundary conditions. The global existence versus blow‐up criteria are studied in terms of the variable exponents. We proposed that the maximums of variable exponents are the key clue to determine blow‐up classifications and describe blow‐up rates for positive solutions. Copyright © 2011 John Wiley & Sons, Ltd.     

Global solutions and blow‐up problems to a porous medium system with nonlocal sources and nonlocal boundary conditions

This paper deals with a porous medium system with nonlocal sources and weighted nonlocal boundary conditions. The main aim of this paper is to study how the reaction terms, the diffusion terms, and the weight functions in the boundary conditions affect the global and blow‐up properties to a porous medium system. The conditions on the global existence and blow‐up in finite time for nonnegative solutions are given. Furthermore, the blow‐up rate estimates of the blow‐up solutions are also established. Copyright © 2011 John Wiley & Sons, Ltd.     

Sign‐changing bubble for Neumann problem with critical nonlinearity on the boundary

The main purpose of this paper is to construct sign‐changing solution for the following Neumann problem: where n≥3 and K is a bounded and continuous function on , which concentrate around two critical points satisfying some conditions. Copyright © 2016 John Wiley & Sons, Ltd.     

带有变指数拟线性椭圆方程组的边界爆破解

研究了在光滑有界域中带有变指数的拟线性椭圆方程组,且该方程组满足边界爆破的条件,在常指数的基础上进一步深入讨论了变指数的情况.主要运用了构造上下解和迭代的方法证明了边界爆破解在临界与次临界条件下,解的存在性,唯一性以及边界行为.     

Sign changing solutions to a nonlinear elliptic problem involving the critical Sobolev exponent in pierced domains

We consider the problem in Ω_ε, u=0 on ∂Ω_ε, where Ω_ε:=ΩB(0,ε) and Ω is a bounded smooth domain in , which contains the origin and is symmetric with respect to the origin, N3 and ε is a positive parameter. As ε goes to zero, we construct sign changing solutions with multiple blow up at the origin.     

Time‐weighted blow‐up rates and pointwise profile for single‐point blow‐up solutions in reaction–diffusion equations

This paper deals with asymptotic behavior for blow‐up solutions to time‐weighted reaction–diffusion equations u_t=Δu+e^αtv^p and v_t=Δv+e^βtu^q, subject to homogeneous Dirichlet boundary. The time‐weighted blow‐up rates are defined and obtained by ways of the scaling or auxiliary‐function methods for all α, . Aiding by key inequalities between components of solutions, we give lower pointwise blow‐up profiles for single‐point blow‐up solutions. We also study the solutions of the system with variable exponents instead of constant ones, where blow‐up rates and new blow‐up versus global existence criteria are obtained. Time‐weighted functions influence critical Fujita exponent, critical Fujita coefficient and formulae of blow‐up rates, but they do not limit the order of time‐weighted blow‐up rates and pointwise profile near blow‐up time. Copyright © 2017 John Wiley & Sons, Ltd.     

Global existence and blow‐up of the solutions for the multidimensional generalized Boussinesq equation

In this paper, the existence and the uniqueness of the global solution for the Cauchy problem of the multidimensional generalized Boussinesq equation are obtained. Furthermore, the blow‐up of the solution for the Cauchy problem of the generalized Boussinesq equation is proved. Copyright © 2007 John Wiley & Sons, Ltd.     

Local well‐posedness and blow‐up criteria of solutions for a rod equation

In this paper we consider a new rod equation derived recently by Dai [Acta Mech. 127 No. 1–4, 193–207 (1998)] for a compressible hyperelastic material. We establish local well‐posedness for regular initial data and explore various sufficient conditions of the initial data which guarantee the blow‐up in finite time both for periodic and non‐periodic case. Moreover, the blow‐up time and blow‐up rate are given explicitly. Some interesting examples are given also. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Blow‐up of solutions for a viscoelastic wave equation with variable exponents

In this paper, we consider a viscoelastic wave equation with variable exponents: where the exponents of nonlinearity p(·) and m(·) are given functions and a,b > 0 are constants. For nonincreasing positive function g, we prove the blow‐up result for the solutions with positive initial energy as well as nonpositive initial energy. We extend the previous blow‐up results to a viscoelastic wave equation with variable exponents.     

Multiple sign‐changing solutions for Kirchhoff‐type equations in

In this paper, we study the following Kirchhoff‐type equations where a>0,b⩾0,4<p<2^∗=6, and . Under some suitable conditions, we prove that the equation has three solutions of mountain pass type: one positive, one negative, and sign‐changing. Furthermore, this problem has infinitely many sign‐changing solutions. Copyright © 2017 John Wiley & Sons, Ltd.     

Multiple sign‐changing solutions for a class of quasilinear elliptic equations

In this paper, we study the following quasilinear Schrödinger equations: where Ω is a bounded smooth domain of , . Under some suitable conditions, we prove that this equation has three solutions of mountain pass type: one positive, one negative, and sign‐changing. Furthermore, if g is odd with respect to its second variable, this problem has infinitely many sign‐changing solutions. Copyright © 2016 John Wiley & Sons, Ltd.     

Multiple solutions to the pure critical exponent problem in domains with a hole of arbitrary size

We consider the problem \(-\Delta u = \left\vert u\right\vert ^{2^\ast-2} u\,{\rm in}\,\Omega, \quad u = 0\,{\rm on}\,\partial\Omega,\) where Ω is a bounded smooth domain in \(\mathbb{R}^{N}\), N ≥ q3, and \(2^{\ast}=\frac{2N}{N-2}\) is the critical Sobolev exponent. We assume that Ω is annular shaped, i.e. there are constants R ₂ >  R ₁ >  0 such that \(\{x \in \mathbb{R}^{N} : R_{1} < |x| < R_{2}\} \subset \Omega\) and \(0 \not\in \Omega.\) We also assume that Ω is invariant under a group Γ of orthogonal transformations of \(\mathbb{R}^{N}\) without fixed points. We establish the existence of multiple sign changing solutions if, either Γ is arbitrary and R ₁/R ₂ is small enough, or R ₁/R ₂ is arbitrary and the minimal Γ-orbit of Ω is large enough. We believe this is the first existence result for sign changing solutions in domains with holes of arbitrary size. The proof takes advantage of the invariance of this problem under the group of Möbius transformations.     

Existence and multiplicity of positive solutions for a class of singular Kirchhoff type problems with sign‐changing potential

In this paper, the existence and multiplicity of positive solutions are obtained for a class of Kirchhoff type problems with two singular terms and sign‐changing potential by the Nehari method.     

Global and blow‐up solutions for non‐linear degenerate parabolic systems

In this paper the degenerate parabolic system u_t=u(u_xx+av). vt=v(v_xx+bu) with Dirichlet boundary condition is studied. For , the global existence and the asymptotic behaviour (α₁=α₂) of solution are analysed. For , the blow‐up time, blow‐up rate and blow‐up set of blow‐up solution are estimated and the asymptotic behaviour of solution near the blow‐up time is discussed by using the ‘energy’ method. Copyright © 2003 John Wiley & Sons, Ltd.     

Global and blow‐up of solutions for a quasilinear parabolic system with viscoelastic and source terms

In this work, we consider an initial boundary value problem related to the quasilinear parabolic equation for m ≥ 2,p ≥ 2, A(t) a bounded and positive definite matrix, and g a continuously differentiable decaying function, and prove, under suitable conditions on g and p, a general decay of the energy function for the global solution and a blow‐up result for the solution with both positive and negative initial energy. Copyright © 2013 John Wiley & Sons, Ltd.     

Numerical blow‐up for the porous medium equation with a source

We study numerical approximations of positive solutions of the porous medium equation with a nonlinear source, where m > 1, p > 0 and L > 0 are parameters. We describe in terms of p, m, and L when solutions of a semidiscretization in space exist globally in time and when they blow up in a finite time. We also find the blow‐up rates and the blow‐up sets, proving that there is no regional blow‐up for the numerical scheme. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004     

On the decay and blow‐up of solution for coupled nonlinear wave equations with nonlinear damping and source terms

In this work, we consider a nonlinear coupled wave equations with initial‐boundary value conditions and nonlinear damping and source terms. Under suitable assumptions on the damping terms and source terms and initial data in the stable set, we obtain that the decay estimates of the energy function is exponential or polynomial by using Nakao's method. By using the energy method, we obtain the blow‐up result of solution with some positive or nonpositive initial energy. Copyright © 2016 John Wiley & Sons, Ltd.     

Strong solutions to the Keller‐Segel system with the weak Ln /2 initial data and its application to the blow‐up rate

We shall show an exact time interval for the existence of local strong solutions to the Keller‐Segel system with the initial data u₀ in L^{n /2}_w (?ⁿ), the weak L^{n /2}‐space on ?ⁿ. If ‖u₀‖ is sufficiently small, then our solution exists globally in time. Our motivation to construct solutions in L^{n /2}_w (?ⁿ) stems from obtaining a self‐similar solution which does not belong to any usual L^p(?ⁿ). Furthermore, the characterization of local existence of solutions gives us an explicit blow‐up rate of ‖u (t)‖ for n /2 < p < ∞ as t → T_max, where T_max denotes the maximal existence time (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A note on blow‐up for fast diffusive p‐Laplacian with sources

In this note we improve the result of Theorem 3.1 in Yin and Jin (Math. Meth. Appl. Sci. 2007; 30 (10):1147–1167) and establish a blow‐up result for certain solution with positive initial energy. Copyright © 2008 John Wiley & Sons, Ltd.     

Existence,uniqueness, and blow‐up rate of large solutions to equations involving the Laplacian on the half line

This paper shows the existence and the uniqueness of the nonnegative viscosity solution of the singular boundary value problem for t >0, , where f is a continuous non‐decreasing function such that f (0)?0, and h is a nonnegative function satisfying the Keller–Osserman condition. Moreover, when h (u )=u ^p with p >3, we obtain the global estimates for the classic solution u (t ) and the exact blow‐up rate of it at t =0. Copyright © 2017 John Wiley & Sons, Ltd.