Arbitrary Positive Initial Energy Research Articles

A Blow-up Result with Arbitrary Positive Initial Energy for Nonlinear Wave Equations with Degenerate Damping Terms

A Blow-up Result with Arbitrary Positive Initial Energy for Nonlinear Wave Equations with Degenerate Damping Terms

On potential wells to a semilinear hyperbolic equation with damping and conical singularity

This article deals with a semilinear hyperbolic equation with damping and conical singularity that was built by Alimohammady et al. (2017) [1], where the weak solution with low initial energy (I(0)<d,d is the potential well depth) was considered. We extend the previous results in the following three aspects. Firstly, we consider the vacuum isolating behavior of the solution for initial energy I(0)≤0 and 0<I(0)<d, respectively. We find that there are two explicit vacuum regions: an annulus and a ball. Moreover, we obtain the asymptotic behavior of the energy functional as t tends to the maximal existence time, and then two necessary and sufficient conditions for the weak solution existing globally and blowing up in finite time are obtained. Secondly, we discuss the weak solution with critical initial energy and establish the global existence and nonexistence results. Finally, the weak solution with arbitrary positive initial energy is studied. In this case, the initial conditions such that the weak solution exists globally and blows up in finite time are given.

Global existence and blow up of solution for semilinear hyperbolic equation with logarithmic nonlinearity

In this paper we consider the semilinear wave equation with logarithmic nonlinearity. By modifying and using potential well combined with logarithmic Sobolev inequality, we derive the global existence and infinite time blow up of the solution at low energy level E(0)<d . Then these results are extended in parallel to the critical case E(0)=d. Besides, with additional assumptions on initial data, the infinite time blow up result is given with arbitrary positive initial energy E(0)>0.

General Decay and Blow-Up Results for Nonlinear Fourth-Order Integro-Differential Equation

This study aims at considering an initial-boundary value problem for nonlinear fourth-order viscoelastic equation in a bounded domain. Under suitable conditions of the initial data and of the relaxation function, it is proved that the solution energy is generally decayed. It is also shown that regarding arbitrary positive initial energy, certain solutions blow-up in a finite time.

A Class of Fourth Order Damped Wave Equations with Arbitrary Positive Initial Energy

AbstractIn this paper, we study the initial boundary value problem for a class of fourth order damped wave equations with arbitrary positive initial energy. In the framework of the energy method, we further exploit the properties of the Nehari functional. Finally, the global existence and finite time blow-up of solutions are obtained.

On behaviour of solutions for a nonlinear viscoelastic equation with variable-exponent nonlinearities

This paper deals with a viscoelastic wave equation with variable-exponent nonlinearities, subject to nonlinear boundary feedback. Under appropriate conditions, a general decay result associated to solution energy is proved. It is also shown that regarding arbitrary positive initial energy, solutions blow-up in a finite time.

Solution Blow-up in a Nonlinear System of Equations with Positive Energy in Field Theory

A problem for a nonlinear system of electromagnetic equations in the Coulomb calibration with allowance for sources of free-charge currents is considered. The local-in-time solvability in the weak sense of the corresponding initial–boundary value problem is proved by applying the method of a priori estimates in conjunction with the Galerkin method. A modified Levine method is used to prove that, for an arbitrary positive initial energy, under a certain initial condition on the functional $$\Phi (t) = \int\limits_\Omega {|A{|^2}dx} $$ , where A(x) is a vector potential, the solution of the initial–boundary value problem blows up in finite time. An upper bound for the blow-up time is obtained.

Non-existence of global solutions to nonlinear wave equations with positive initial energy

We consider the Cauchy problem for nonlinear abstract wave equations in a Hilbert space. Our main goal is to show that this problem has solutions with arbitrary positive initial energy that blow up in a finite time. The main theorem is proved by employing a result on growth of solutions of abstract nonlinear wave equation and the concavity method. A number of examples of nonlinear wave equations are given. A result on blow up of solutions with arbitrary positive initial energy to the initial boundary value problem for the wave equation under nonlinear boundary conditions is also obtained.

A blow-up result for a nonlinear wave equation with variable-exponent nonlinearities

In this paper, we consider the following wave equation:in which is a nonnegative constant and the exponents of nonlinearity are given functions. For arbitrary positive initial energy, we prove a finite-time blow-up result.

Revised concavity method and application to Klein-Gordon equation

A revised version of the concavity method of Levine, based on a new ordinary differential inequality, is proposed. Necessary and sufficient condition for nonexistence of global solutions of the inequality is proved. As an application, finite time blow up of the solution to Klein-Gordon equation with arbitrary positive initial energy is obtained under very general structural conditions.

Blow‐up phenomena for a class of fourth‐order nonlinear wave equations with a viscous damping term

This paper deals with the blow‐up phenomena for a class of fourth‐order nonlinear wave equations with a viscous damping term urn:x-wiley:mma:media:mma3623:mma3623-math-0046 with Ω = (0,1) and α &gt; 0. Here, f(s) is a given nonlinear smooth function.For 0 &lt; α &lt; p – 1, we prove that the blow‐up occurs in finite time for arbitrary positive initial energy and suitable initial data. This result extends the recent results obtained by Xu et al. (Applicable Analysis)(2013) and Chen and Lu (J. Math. Anal. Appl.)(2009). Copyright © 2015 John Wiley &amp; Sons, Ltd.

Blow-up of solution for an integro-differential equation with arbitrary positive initial energy

In this paper, we consider the integro-differential equation $u_{tt}-M(\|\nabla u\|^{2}_{2})\Delta u+ \int_{0}^{t} g(t-\tau)\Delta u(\tau)\, d\tau+ u_{t}=f(u)$ , $(x,t)\in\Omega\times(0,T)$ , with initial and Dirichlet boundary conditions. Under suitable assumptions on the functions g and the initial data, a blow-up result with arbitrary positive initial energy is established.

Sign-preserving functionals and blow-up to Klein–Gordon equation with arbitrary high energy

The global behaviour of the weak solutions of the Cauchy problem to nonlinear Klein–Gordon equation with combined power-type nonlinearity is studied. Finite time blow-up of the solutions with arbitrary high positive initial energy is proved under general structural conditions on the initial data. A new functional, invariant under the flow of the equation, is introduced and investigated.

Blow-up in a semilinear parabolic problem with variable source under positive initial energy

In this paper, we consider a class of semilinear parabolic equations with variable source under homogeneous Dirichlet boundary conditions in a bounded domain with smooth boundary and For all we prove that the classical solutions to the above equation blow-up in finite time with arbitrary positive initial energy and suitable large initial values. This result extends a recent result by Wu et al. [Appl. Math. Lett. 2013;26:539–543] which asserts the blow-up of solutions for , provided that

Nonlinear wave equations and reaction-diffusion equations with several nonlinear source terms of different signs at high energy level

This paper is concerned with the initial boundary value problem of a class of nonlinear wave equations and reaction–diffusion equations with several nonlinear source terms of different signs. For the initial boundary value problem of the nonlinear wave equations, we derive a blow up result for certain initial data with arbitrary positive initial energy. For the initial boundary value problem of the nonlinear reaction–diffusion equations, we discuss some probabilities of the existence and nonexistence of global solutions and give some sufficient conditions for the global and nonglobal existence of solutions at high initial energy level by employing the comparison principle and variational methods. doi:10.1017/S1446181113000175

Global solutions and finite time blow up for damped Klein-Gordon equation

We study the Cauchy problem of strongly damped Klein-Gordon equation. Global existence and asymptotic behavior of solutions with initial data in the potential well are derived. Moreover, not only does finite time blow up with initial data in the unstable set is proved, but also blow up results with arbitrary positive initial energy are obtained.

A blow-up result in a system of nonlinear viscoelastic wave equations with arbitrary positive initial energy

In this paper we consider a system of nonlinear viscoelastic wave equations. Under arbitrary positive initial energy and standard conditions on the relaxation functions, we prove a finite-time blow-up result.

NONLINEAR WAVE EQUATIONS AND REACTION–DIFFUSION EQUATIONS WITH SEVERAL NONLINEAR SOURCE TERMS OF DIFFERENT SIGNS AT HIGH ENERGY LEVEL

AbstractThis paper is concerned with the initial boundary value problem of a class of nonlinear wave equations and reaction–diffusion equations with several nonlinear source terms of different signs. For the initial boundary value problem of the nonlinear wave equations, we derive a blow up result for certain initial data with arbitrary positive initial energy. For the initial boundary value problem of the nonlinear reaction–diffusion equations, we discuss some probabilities of the existence and nonexistence of global solutions and give some sufficient conditions for the global and nonglobal existence of solutions at high initial energy level by employing the comparison principle and variational methods.

Global well-posedness for strongly damped viscoelastic wave equation

We study the initial boundary value problem for the nonlinear viscoelastic wave equation with strong damping term and dispersive term. By introducing a family of potential wells we not only obtain the invariant sets, but also prove the existence and nonexistence of global weak solution under some conditions with low initial energy. Furthermore, we establish a blow-up result for certain solutions with arbitrary positive initial energy (high energy case)

FINITE TIME BLOW-UP FOR THE NONLINEAR FOURTH-ORDER DISPERSIVE-DISSIPATIVE WAVE EQUATION AT HIGH ENERGY LEVEL

In this paper, we investigate the initial boundary value problem of the nonlinear fourth-order dispersive-dissipative wave equation. By using the concavity method, we establish a blow-up result for certain solutions with arbitrary positive initial energy.