The Long Time Behavior of Solutions for Coupled Equations 

Author  HuZuoYao 
Tutor  WangZeJia 
School  Jilin University 
Course  Applied Mathematics 
Keywords  Coupled equations Global existence Blow up Critical curve 
CLC  O175.26 
Type  Master's thesis 
Year  2010 
Downloads  11 
Quotes  0 
In This paper we introduce the long time behavior of solutions for various diffusion systems with sources and discuss global existence of the solutions, blow up time of solutions, the critical curve for systems, blowup rate estimates for systems and blowup sets for systems.In the introduction section, first of all,we described based content and applications of the blow up for coupled equations, and briefly describes the important research results based on the blow up for diffusion equation in recent years.In the first chapter we present the long time behavior of solutions for linear diffusion equations. According to the location of sources, we describe the asymptotic behavior of solutions for coupled equations with interior flux and the asymptotic behavior of solutions for coupled equations with boundary flux. We also have a detailed description of the problems such as global existence of solutions, the blow up of the solutions and critical exponents for systems.Escobedo and Herrero researched the linear diffusion equations with interior sources the solutions of (1) with CauchyDirichlet problem have the following properties:(i)If 0<pq≤1, every solution of (1) is global in time. (ⅱ) If pq>1, then the solution of (1) blows up in finite time for sufficiently large initial values and exists globally for sufficiently small initial values.For the linear diffusion equations with nonlocal sources the solutions of (2) with CauchyDirichlet problem have the following properties:(ⅰ)If 0<pq≤1, every solution of (2) is global in time. (ⅱ) If pq>1, then the solution of (2) blows up in finite time for sufficiently large initial values and exists globally for sufficiently small initial values.Deng, Fila and Levine study the large time behavior of nonnegative solutions of a system as follows: where p, q>0, both u0(x) andν0(x) are nonnegative bounded functions satisfying the compatibility condition. Set, when pq≠1, the solutions of (3) have the following properties:(i)If pq≤1 all nonnegative solutions of (3) are global. (ⅱ) If pq>1, then there are no nontrivial global nonnegative solutions of (3) if max{α,β}≥(?)while both global nontrivial and nonglobal solutions exist if max{α,β}<(?).In the second chapter we present the long time behavior of solutions for nonlinear diffusion equations. According to the location of sources, we describe the asymptotic behavior of solutions for coupled equations with interior flux and the asymptotic behavior of solutions for coupled equations with boundary flux. We also have a detailed description of the problems such as global existence of solutions, the blow up of the solutions and critical exponents for systems.For the nonlinear diffusion equations with interior sources the solutions of the cauchy problem (4) have the following properties: when p, q≥1, if pq<αβ+2 max(β+p,α+q)/n, then the solutions of (4) blow up in finite time; if pq>αβ+2 max(β+p,α+q)/n, then the cauchy problem (4) has both nonglobal and nontrivial global solutions. For the nonlinear diffusion equations with nonlocal sources the solutions of (5) with CauchyDirichlet problem have the following properties:(ⅰ)If m>p1,n>P2 and q1q2<(mp1)(np2), then every nonnegative solution of (5) is global. (ⅱ)If m< p1or n< p2 or q1q2> (mp1)(np2), then the nonnegative solution of (5) blows up in finite time for sufficiently large initial values and exists globally for sufficiently small initial values. (ⅲ) If m> p1, n> P2 and q1q2=(mp1)(nP2), then the nonnegative solutions of (5) exist globally for sufficiently the measure of the domain(│Ω│). Quiros and Rossi considered the nonlinear diffusion equations with boundary sources the solutions of(6)have the following properties:(i)If pq≤((m+ 1)/2)((n+1)/2),every nonnegative solution of(6)is global in time. (ii)When pg>((m+1)/2)((n+1)/2),ifα1+β1≤0 orα2+β2≤0, then every nonnegative,nontrivial solution blow up in finite time；ifα1+β1>0,α2+β2>0,there exist nonnegative solutions with blowup and nonnegative that are global.αi+βi=0(i=1,2)is the critical Fujita type curve of(6).