Blow-up Properties Of Solutions Research Articles

Blowup property of solutions in the parabolic equation with p-Laplacian operator and multi-nonlinearities

ABSTRACT In this paper, we study blowup properties of weak solutions in their norm to the degenerate parabolic equation with multi-nonlinearities and gradient terms. First, we show the existence and uniqueness of weak solutions by using the priori estimate methods. Second, we obtain the global existence criteria and blowup criteria after proving some gradient estimates for different coefficients. Third, we use some Sobolev's inequalities and deal with some differential inequalities of new barrier functions to determine some upper and lower bounds of blowup time of solutions. It could be found out that the blowup or global existence phenomena depend sensitively on the relationship among the different exponents of nonlinearities, which discover a key clue to the different effect of diffusion, gradient term, source term and absorption term on the singular properties of weak solutions.

Time-Space Fractional Diffusion Problems: Existence, Decay Estimates and Blow-Up of Solutions

The aim of this paper is to study the following time-space fractional diffusion problem ∂tβu+(-Δ)αu+(-Δ)α∂tβu=λf(x,u)+g(x,t)inΩ×R+,u(x,t)=0in(RN\\Ω)×R+,u(x,0)=u0(x)inΩ,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} \\displaystyle \\partial _t^\\beta u+(-\\Delta )^\\alpha u+(-\\Delta )^\\alpha \\partial _t^\\beta u=\\lambda f(x,u) +g(x,t) &{}\ ext{ in } \\Omega \ imes {\\mathbb {R}}^{+},\\\\ u(x,t)=0\\ \\ &{}\ ext{ in } ({\\mathbb {R}}^N{\\setminus }\\Omega )\ imes {\\mathbb {R}}^+,\\\\ u(x,0)=u_0(x)\\ &{}\ ext{ in } \\Omega ,\\\\ \\end{array}\\right. } \\end{aligned}$$\\end{document} where Omega subset {mathbb {R}}^N is a bounded domain with Lipschitz boundary, (-Delta )^{alpha } is the fractional Laplace operator with 0<alpha <1, partial _t^{beta } is the Riemann-Liouville time fractional derivative with 0<beta <1, lambda is a positive parameter, f:Omega times {mathbb {R}}rightarrow {mathbb {R}} is a continuous function, and gin L^2(0,infty ;L^2(Omega )). Under natural assumptions, the global and local existence of solutions are obtained by applying the Galerkin method. Then, by virtue of a differential inequality technique, we give a decay estimate of solutions. Moreover, the blow-up property of solutions is also investigated.

Blowup Phenomenon of Solutions for the IBVP of the Compressible Euler Equations in Spherical Symmetry.

The blowup phenomenon of solutions is investigated for the initial-boundary value problem (IBVP) of the N-dimensional Euler equations with spherical symmetry. We first show that there are only trivial solutions when the velocity is of the form c(t)|x|α−1 x + b(t)(x/|x|) for any value of α ≠ 1 or any positive integer N ≠ 1. Then, we show that blowup phenomenon occurs when α = N = 1 and . As a corollary, the blowup properties of solutions with velocity of the form are obtained. Our analysis includes both the isentropic case (γ > 1) and the isothermal case (γ = 1).

Blow-up for a parabolic system with nonlocal sources and nonlocal boundary conditions

This paper deals with blow-up properties of solutions to a nonlocal parabolic system with nonlocal boundary conditions. The global existence and finite time blow-up criteria are obtained. Moreover, for some special cases, we establish the precise blow-up rate estimates.

Global existence and blow-up properties of solutions for porous medium equation with nonlinear memory and weighted nonlocal boundary condition

In this paper, we investigate a initial boundary value problem of porous medium equation with nonlocal boundary condition. We find the effects of weight function in the boundary condition and competitive relationship between nonlinear memory term and inner absorption term on whether determining blow-up of solutions or not, and give the blow-up rate estimate under some suitable condition.

Existence and Blow-Up Properties of Solutions for a Porous Medium Equation

In this paper, a porous medium equation with nonlocal boundary condition and a localized source is considered. The existence and blow-up properties of solutions are obtained under some conditions.

A porous medium equation with nonlocal boundary condition and a localized source

This article studies the blow-up properties of solutions to a porous medium equation with nonlocal boundary condition and a general localized source. Conditions for the existence of global or blow-up solutions are obtained. Moreover, it is proved that the unique solution has global blow-up property whenever blow-up occurs. Blow-up rate estimates are also obtained for some special cases.

Non-simultaneous blow-up in localized reaction–diffusion equations

This article deals with blow-up solutions in reaction–diffusion equations coupled via localized exponential sources, subject to null Dirichlet conditions. The optimal and complete classification is obtained for simultaneous and non-simultaneous blow-up solutions. Moreover, blow-up rates and blow-up sets are also discussed. It is interesting that, in some exponent regions, blow-up phenomena depend sensitively on the choosing of initial data, and the localized nonlinearities play important roles in the blow-up properties of solutions.

The blow-up rate for a system of heat equations with neumann boundary conditions

This paper deals with the blow-up properties of solutions to a system of heat equations ut=Δu, vt=Δv in BR×(0, T) with the Neumann boundary conditions eu/eη=ev, ev/eη=eu on SR×[0, T). The exact blow-up rates are established. It is also proved that the blow-up will occur only on the boundary.

On critical exponents for semilinear heat equations with nonlinear boundary conditions

This paper deals with the blow-up properties of solutions to semilinear heat equation \(u_t - \Delta u = u^p {\text{ in }}R_ + ^N {\text{ }}x (0,T)\) with the nonlinear boundary condition \( - \frac{{\partial u}}{{\partial x}} = u^q for{\text{ }}x1 = t \in (T)\). It has been proved that if max(p,q)≤1,every nonnegative solution is global. When min(p, q)>1 by letting α=1/p−1 and β=1/2(q−1) it follows that if max (α,β)≤N/2,all nontrivial non-negative solutions are nonglobal, whereas if max(α,β)< N/2,there exist both global and non-global solutions. Moreover, the exact blow-up rates are established.

The blow-up property for a system of heat equations with nonlinear boundary conditions

This paper deals with the blow-up properties of solutions to the systems u t=Δu,vt=Δv in B RX(0,T) subject to nonlinear boundary conditions $$\frac{{\partial u}}{{\partial \eta }} = \upsilon ^p ,\frac{{\partial \upsilon }}{{\partial \eta }} = u^q $$ , in S RX(0,T). It is shown that under certain conditions the solution blows up at a finite time and the blow-up only occurs on the boundary. The self-similar solution for the one-dimensional case has been studied. Moreover, the exact blow-up rates are also derived.

The Asymptotic Behaviour and Blow-up Properties of Solutions of a Non-local Burgers' Equation

In this paper, we continue the work of [1] and obtain a complete picture of asymptotic behavior and blow-up properties of solutions of a non-local Burgers' equation.