Existence Theorem Of Global Solutions Research Articles

On the existence with exponential decay and the blow-up of solutions for coupled systems of semi-linear corner-degenerate parabolic equations with singular potentials

In this article, we study the initial boundary value problem of coupled semi-linear degenerate parabolic equations with a singular potential term on manifolds with corner singularities. Firstly, we introduce the corner type weighted p-Sobolev spaces and the weighted corner type Sobolev inequality, the Poincare inequality, and the Hardy inequality. Then, by using the potential well method and the inequality mentioned above, we obtain an existence theorem of global solutions with exponential decay and show the blow-up in finite time of solutions for both cases with low initial energy and critical initial energy. Significantly, the relation between the above two phenomena is derived as a sharp condition. Moreover, we show that the global existence also holds for the case of a potential well family.

Smoothing effects of the initial-boundary value problem for logarithmic type quasilinear parabolic equations

We give existence theorems of global solutions in Lloc∞((0,∞);W01,∞) to the initial boundary value problem for quasilinear degenerate parabolic equations of the form ut−div{σ(|∇u|2)∇u}=0, where the class of σ(v2) includes the logarithmic case σ(|∇u|2)= log (1+|∇u|2) for a typical example. We assume that the initial data belong to W01,p0,p0≥2, or Lr,r≥1, and we derive precise estimates for ‖∇u(t)‖∞ near t=0.

Global Existence and Blow-Up of Solutions for Nonlinear Klein-Gordon Equation with Damping Term and Nonnegative Potentials

This paper is concerned with the nonlinear Klein-Gordon equation with damping term and nonnegative potentials. We introduce a family of potential wells and discuss the invariant sets and vacuum isolating behavior of solutions. Using the potential well argument, we obtain a new existence theorem of global solutions and a blow-up result for solutions in finite time.

Global solutions to a chemotaxis system with non-diffusive memory

In this article, an existence theorem of global solutions with small initial data belonging to L1∩Lp, (n<p⩽∞) for a chemotaxis system is given on the whole space Rn, n⩾3. In the case p=∞, our global solution is integrable with respect to the space variable on some time interval, and then conserves the mass for a short time, at least. The system consists of a chemotaxis equation with a logarithmic term and an ordinary equation without diffusion term.

Global existence and nonexistence for semilinear parabolic equations with conical degeneration

In this article we study the initial boundary value problem of semilinear parabolic equations \({u_t-\triangle_\mathbb{B}u=|u|^{p-1}u}\) on a manifold with conical singularity, where \({\triangle_\mathbb{B}}\) is Fuchsian type Laplace operator investigated in Chen et al. (Calc Var 43:463–484, 2012) with totally characteristic degeneracy on the boundary x1 = 0. By using a family of potential wells, we obtain existence theorem of global solutions with exponential decay and show the blow-up in finite time of solutions. Especially, the relation between the above two phenomena is derived as a sharp condition.

Sharp Condition for Global Existence and Blow‐Up on Klein‐Gordon Equation

We study the initial boundary value problem of the nonlinear Klein‐Gordon equation. First we introduce a family of potential wells. By using them, we obtain a new existence theorem of global solutions and show the blow‐up in finite time of solutions. Especially the relation between the above two phenomena is derived as a sharp condition.

Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces

In this paper, first a new fixed point theorem is established, and then, by the use of it, the existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces are investigated. The results obtained in this paper generalize and improve the results corresponding to those obtained by others.

Blow-up solutions and global solutions for a class of quasilinear parabolic equations with robin boundary conditions

The type of problem under consideration is u t = ∇ ⋅ ( a ( u ) b ( x ) ∇ u ) + g ( x , t ) f ( u ) , i n D × ( 0 , T ) , ∂ u ∂ n + σ ( x , t ) u = 0 , o n ∂ D × ( 0 , T ) , u ( x , 0 ) = u 0 ( x ) > 0 , i n D _ where D is a smooth bounded domain of R N . By constructing an auxiliary function and using Hopf's maximum principles on it, existence theorems of blow-up solutions, upper bound of ‘blow-up time”, upper estimates of “blow-up rate”, existence theorems of global solutions and upper estimates of global solutions are given under suitable assumptions on a, b, f, g, σ, and initial data u 0( x). The obtained results are applied to some examples in which a, b, f, g, and σ are power functions or exponential functions.

Existence theorems of global solutions of initial value problems for nonlinear integrodifferential equations of mixed type in banach spaces and applications

This paper investigates the existence of global solutions for initial value problems of the nonlinear integrodifferential equations in Banach spaces. We obtain the existence of global solutions by Schauder's fixed-point theorem. Our results are the generalizations and improvements of recent results. As applications of our results, the existence of global solutions of two classes of third-order mixed boundary value problem is obtained, and an example of infinite system for nonlinear integrodifferential equations is worked out to illustrate our main results.

Multidimensional viscoelasticity equations with nonlinear damping and source terms

In this paper we study the initial boundary value problem of multidimensional viscoelasticity equations u tt− Δu t− ∑ i=1 N ∂ ∂x i σ i(u x i )+f(u t)=g(u), x∈Ω, t>0, u(x,0)=u 0(x), u t(x,0)=u 1(x), x∈Ω, u(x,t)=0, x∈∂Ω, t⩾0, where Ω⊂ R N is a bounded domain, ( σ i ( s 1)− σ i ( s 2))( s 1− s 2)>0, f( s) s⩾0, g( s) s⩾0 and satisfy some growth conditions. First we introduce a family of potential wells by using a new method. Then using it we obtain some new existence theorems of global solutions and behaviour of vacuum isolation of solutions.

On potential wells and vacuum isolating of solutions for semilinear wave equations

In this paper, we study the initial boundary value problem of semilinear wave equations: utt−Δu=|u|p−1u,x∈Ω,t>0,u(x,0)=u0(x),ut(x,0)=u1(x),x∈Ω,u(x,t)=0,x∈∂Ω,t⩾0,where Ω⊂RN is a bounded domain, 1<p<∞ for N=1,2; 1<p⩽N+2N−2 for N⩾3. First, by using a new method, we introduce a family of potential wells which include the known potential well as a special case. Then by using it, we obtain some new existence theorems of global solutions, and prove that for any e∈(0,d) (d is the depth of the known potential well) all solutions with initial energy E(0) satisfying 0<E(0)⩽e can only lie either inside of some smaller ball or outside of some bigger ball of space H01(Ω).

Global Solutions of Nonlinear Magnetohydrodynamics with Large Initial Data

A free boundary problem for nonlinear magnetohydrodynamics with general large initial data is investigated. The existence, uniqueness, and regularity of global solutions are established with large initial data in H1. It is shown that neither shock waves nor vacuum and concentration in the solutions are developed in a finite time, although there is a complex interaction between the hydrodynamic and magnetodynamic effects. An existence theorem of global solutions with large discontinuous initial data is also established.

Hyperbolic systems of conservation laws with a symmetry

Hyperbolic systems of conservaiton laws with a symmetry are studied. Some peculiar phenomena for such systems are shown. Admissibility criteria for solutions to such systems are discussed. propagation and cancellation of initial oscillations for the systems are classified. As a byproduct of this study, an existence theorem of global solutions for the Cauchy of the systems is established.

Convergence of the fractional step Lax-Friedrichs scheme and Godunov scheme for the isentropic system of gas dynamics

A convergence theorem of the fractional step Lax-Friedrichs scheme and Godunov scheme for an inhomogeneous system of isentropic gas dynamics (1<γ≦5/3) is established by using the framework of compensated compactness. Meanwhile, a corresponding existence theorem of global solutions with large data containing the vacuum is obtained.