Blow-up Of Solutions For Some Systems Of Heat Equations With Nonlinear Boundary Conditions

This article is a survey. Blow-up properties of solutions forheat equations with nonlinear boundary conditions are studied. Theblow-up theory of solutions to nonlinear equations is an importantaspect to partial di?erential equations. The problems come fromphysics,chemistry,medicine,biology and other fields. In this paper,from variety of di?erent types of nonlinear heat equations, we mainlystate the blow-up rate,blow-up set of solutions to three classes ofheat equations. In this paper ,through the discussion of the criticalindex, Using Green's identities, maximum principle and the methodof upper and lower solutions, we give a systematic discussion on theglobal existence and the blowing-up in a finite time and the estimateof blow-up rate to the types of solutions of nonlinear heat equations.In the first chapter, we introduce the issues of the develop-ment process, background, knowledge and theoretical basis of theproblems.In the second chapter, We consider the following heat equationswith nonlinear boundary conditionswhere p,q > 0,0≤α< 1,0≤β< p, is a bounded domainwith smooth boundary andηis the exterior normal vector on . Also, the non-zero and non-negative functions u₀(x) and v₀(x)satisfy the compatibility conditions and . By using Comparison principle, we obtain the results of the globalexistence and the blowing-up in a finite time and the estimate ofblow-up rate. By analyzing the factors in system, especially non-linear boundary value of the role plays in blowing up problems, theblow-up rate under certain conditions was further obtained.Theorem1. The solutions of (1) always blow up in finitetime for non-zero and non-negative initial data. Moreover, u and vblow up simultaneously if p,q > 0, 0≤α< 1, and 0≤β< p ,u₀≥0, and satisfies the compatibility condition for Also, there exist positive constants C₁,C₂,C₃,C₄ suchthatTheorem2. Assume that p,q > 0. u0 0, and satisfiesthe compatibility condition for .Then there exist two positive constants C5,C6 such thatfor any (x,t)∈Ω'×[0,T) with . Here .That is, the blow-up only occurs on the boundary.In the third chapter , we consider the heat equations with the LetΩbe a bounded domain in Rⁿ with smooth boundary , andletηbe the exterior normal vector on .Here constantsα,β≥0and p,q > 0, initial data u₀,v₀ are continuous non-negative andnon-trivial functions, and satisfy the compatibility conditions.Mainly discussing: (1) solution blows up in finite time, andsolution (u,v) simultaneously blows up; (2) the estimate of blow-uprate; (3) blow-up point set of solutions.The estimate of blow-up rate to the problem (3)is following:Theorem3. SupposeΩ= B(0;R). Let the initial datau₀,v₀ be radially symmetric and non-decreasing functions, and letthem satisfyΔu₀,Δv₀≥0. Assume that(H₂) u(x,t),v(x,t)blow-up simultaneously in finite time T.Then the following estimate holds:Moreover, ifα< 1 then the estimatesalso hold. for some positive constants C and c. Theorem4. Let k =α+ p(q + 1 -α)/(p + 1 -β), that isk > 1. If the solution (u,v) satisfys u_t,v_t≥0.ThenIn particular, if q -α= p -β, then, for l =β+ q(p + 1 -β)/(q +1 -α) > 1, (0 < t < T),Theorem5. AssumeΩ= B(0;R).Let u0,v0 be radiallysymmetric functions, and let them satisfy u₀, v₀≥0. Thenthe conclusion of Theorem 3 holds. Moreover,if one of the followingtwo conditions holds:(1)α< 1, orβ< 1;(2)α,β≥1, and q≥1, (u0)_r,( v0)_r≥0.Then (u,v) satisfy the following estimateIn the forth chapter, we consider the following semilinear heatequation with nonlinear nonlocal boundary condition:whereΩis a bounded domain in Rn for n≥1 with a smooth bound-ary ,p > 0 and l > 0. Here c(x,t) is a nonnegative locally Ho¨ldercontinuous function defined for and t≥0 and k(x,y,t) is anonnegative continuous function defined for , and t≥0. The initial datum u0(x) is a nonnegative continuous functionsatisfying the boundary condition at t = 0.By constructing the special bounded supersolution, we obtainthe condition under which the global solution exists.Theorem6. Let max(p,l)≤1,. Then problem (3) has globalsolutions for any coefficients c(x,t)andk(x,y,t),u_o(x) and any nonnegative initial data.For convenience,look at the Cauchy problems:v′(t) = -λ₁v + c₀(t)v^p, p > 1,0 < l < 1,v′(t) = -λ₁v + k₀(t)v^l, 0 < p < 1,l > 1,v′(t) = -λ₁v + c₀(t)v^p + k₀(t)v, p > 1,l = 1, (4)v′(t) = -λ₁v + c₀(t)v + k₀(t)v^l, p = 1,l > 1,v′(t) = -λ₁v + c₀(t)v^p + k₀(t)v^l, p > 1,l > 1.initial dataTheorem7. Let max(p,l)≤1, and the Cauchy problem (4),(5) does not have a global solution. Then the solution of (3) blowsup in a finite time.Corollary1. Note that from Theorem 8, we can get condi-tions for nonexistence of global solutions in the case that p > 1,ifand in the case that l > 1, if In particular, there are not nontrivial nonnegative global solutionsof (3) if p > 1 andTheorem8. Let max(p,l) > 1, and p(u,t)≥δ(t)u^max(p,l),u≥0,t≥0 for any u≥0 and t≥0. Then any solution of (3) withnontrivial initial datum blows up in a finite time.Theorem9. Let min(p,l) > 1, andhold.Then there exist nonnegative solutions of (3) with su?cientlysmall initial data, which are globally bounded.We can get more information for global existence and blow-upin a finite time of solutions for (3) when either p = 1 or l = 1. Thefollowing statement deals with the case that p = 1 with l > 1 andneeds two new assumptions thatwhereβ≥0,M > 0 andγ> 0.Theorem10. Let p = 1 and l > 1.If c(x,t)≥β,k(x,y,t),satisfies (7), then any solution of (3) with nontrivial initial datumblows up in a finite time.On the other hand, if with some constant c and (8) is valid, then there exist nonnegativesolutions of (3) with sufficiently small initial data, which are globallybounded.Remark1. We putγ=λ₁ and suppose that k(x,y,t)≡M0 > 0, . Now we con-sider the Cauchy problemwhere the solution g(t) of (8) can be written asIt is obvious that g(t) blows up in a finite time for allα> 0 if, forexample, Thus, ifω(0)≥α, then by comparison principle for the Cauchyproblem (8)ω(t)≥g(t)., which in turn implies that for anyω(0) >0 functionω(t) exists only finite time ifδ(t) satisfies . Therefore, any solution of (3) with nontrivial initial datumblows up in a finite time.