Blow‐up phenomena for a system of semilinear parabolic equations with nonlinear boundary conditions

This paper deals with the blow‐up phenomena for a system of parabolic equations with nonlinear boundary conditions. We show that under some conditions on the nonlinearities, blow‐up occurs at some finite time. We also obtain upper and lower bounds for the blow‐up time when blow‐up occurs. Copyright © 2014 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)     

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.     

On a reaction diffusion equation with nonlinear time‐nonlocal source term

In this article, we prove the local existence of a unique solution to a nonlocal in time and space evolution equation with a time nonlocal nonlinearity of exponential growth. Moreover, under some suitable conditions on the initial data, it is shown that local solutions experience blow‐up. The time profile of the blowing‐up solutions is also presented. Copyright © 2015 John Wiley & Sons, Ltd.     

Blow‐up profiles of solutions for the exponential reaction‐diffusion equation

We consider the blow‐up of solutions for a semilinear reaction‐diffusion equation with exponential reaction term. It is known that certain solutions that can be continued beyond the blow‐up time possess a non‐constant self‐similar blow‐up profile. Our aim is to find the final time blow‐up profile for such solutions. The proof is based on general ideas using semigroup estimates. The same approach works also for the power nonlinearity. Copyright © 2011 John Wiley & Sons, Ltd.     

Blow‐up phenomena in the model of a space charge stratification in semiconductors: analytical and numerical analysis

The initial‐boundary value problems for a Sobolev equation with exponential nonlinearities, classical, and nonclassical boundary conditions are considered. For this model, which describes processes in crystalline semiconductors, the blow‐up phenomena are studied. The sufficient blow‐up conditions and the blow‐up time are analyzed by the method of the test functions. This analytical a priori information is used in the numerical experiments, which are able to determine the process of the solution's blow‐up more accurately. The model derivation and some questions of local solvability and uniqueness are also discussed. Copyright © 2016 John Wiley & Sons, Ltd.     

On the blow‐up phenomena for a 1‐dimensional equation of ion sound waves in a plasma: Analytical and numerical investigation

The initial‐boundary value problem for an equation of ion sound waves in plasma is considered. A theorem on nonextendable solution is proved. The blow‐up phenomena are studied. The sufficient blow‐up conditions and the blow‐up time are analysed by the method of the test functions. This analytical a priori information is used in the numerical experiments, which are able to determine the process of the solution's blow‐up more accurately.     

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.     

Local and blowing‐up solutions for a space‐time fractional evolution system with nonlinearities of exponential growth

Local and blowing‐up solutions for the Cauchy problem for a system of space and time fractional evolution equations with time‐nonlocal nonlinearities of exponential growth are considered. The existence and uniqueness of the local mild solution is assured by the Banach fixed point principle. Then, we establish a blow‐up result by Pokhozhaev capacity method. Finally, under some suitable conditions, an estimate of the life span of blowing‐up solutions is established.     

Blow‐up phenomena in chemotaxis systems with a source term

This paper deals with a parabolic–parabolic Keller–Segel‐type system in a bounded domain of , {N = 2;3}, under different boundary conditions, with time‐dependent coefficients and a positive source term. The solutions may blow up in finite time t^?; and under appropriate assumptions on data, explicit lower bounds for blow‐up time are obtained when blow up occurs. Copyright © 2015 John Wiley & Sons, Ltd.     

Remarks on the qualitative behavior of the undamped Klein‐Gordon equation

We consider the undamped Klein‐Gordon equation in bounded domains with homogeneous Dirichlet boundary conditions. For any real value of the initial energy, particularly for supercritical values of the energy, we give sufficient conditions to conclude blow‐up in finite time of weak solutions. The success of the analysis is based on a detailed analysis of a differential inequality. Our results improve previous ones in the literature.     

Asymptotic analysis and estimates of blow‐up time for the radial symmetric semilinear heat equation in the open‐spectrum case

We estimate the blow‐up time for the reaction diffusion equation u_t=Δu+ λf(u), for the radial symmetric case, where f is a positive, increasing and convex function growing fast enough at infinity. Here λ>λ^*, where λ^* is the ‘extremal’ (critical) value for λ, such that there exists an ‘extremal’ weak but not a classical steady‐state solution at λ=λ^* with ∥w(?, λ)∥_∞→∞ as 0<λ→λ^*?. Estimates of the blow‐up time are obtained by using comparison methods. Also an asymptotic analysis is applied when f(s)=e^s, for λ?λ^*?1, regarding the form of the solution during blow‐up and an asymptotic estimate of blow‐up time is obtained. Finally, some numerical results are also presented. Copyright © 2007 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     

Wave‐breaking phenomenon for a generalized spatially periodic Camassa–Holm system

Considered herein is a generalized two‐component Camassa–Holm system in spatially periodic setting. We first prove two conservation laws; then under proper assumptions on the initial data, we show the precise blow‐up scenarios and sufficient conditions guaranteeing the formation of singularities to the solutions of the generalized Camassa–Holm system. Copyright © 2014 John Wiley & Sons, Ltd.     

Blow‐up phenomena for a semilinear parabolic equation with weighted inner absorption under nonlinear boundary flux

Blow‐up phenomena for a nonlinear divergence form parabolic equation with weighted inner absorption term are investigated under nonlinear boundary flux in a bounded star‐shaped region. We assume some conditions on weight function and nonlinearities to guarantee that the solution exists globally or blows up at finite time. Moreover, by virtue of the modified differential inequality, upper and lower bounds for the blow‐up time of the solution are derived in higher dimensional spaces. Three examples are presented to illustrate applications of our results. Copyright © 2016 John Wiley & Sons, Ltd.     

Blow‐up analysis for a system of heat equations coupled via nonlinear boundary conditions

In this paper, we study a system of heat equations coupled via nonlinear boundary conditions (1) Here p, q>0. We prove that the solutions always blow up in finite time for non‐trivial and non‐negative initial values. We also prove that the blow‐up occurs only on S_R = ?B_R for Ω = B_R = {x ? ?ⁿ:|x|<R}and under some assumptions on the initial values. Copyright © 2007 John Wiley & Sons, Ltd.     

Qualitative properties of solutions to a nonlocal evolution system

In this paper, we investigate the local existence and the finite‐time blow‐up of solutions for a semi‐linear parabolic system with a nonlinear memory. Moreover, we give the blow‐up rate of solutions and the necessary conditions for local or global existence. Copyright © 2011 John Wiley & Sons, Ltd.     

Local well‐posedness and blow‐up criterion for a compressible Navier‐Stokes‐Fourier‐P1 approximate model arising in radiation hydrodynamics

We establish a local well‐posedness and a blow‐up criterion of strong solutions for the compressible Navier‐Stokes‐Fourier‐P1 approximate model arising in radiation hydrodynamics. For the local well‐posedness result, we do not need the assumption on the positivity of the initial density and it may vanish in an open subset of the domain.     

A finite‐time blow‐up result for a class of solutions with positive initial energy for coupled system of heat equations with memories

In this article, we are interested by a system of heat equations with initial condition and zero Dirichlet boundary conditions. We prove a finite‐time blow‐up result for a large class of solutions with positive initial energy.