Single Point Blow-up For Evolution M-Laplace Equation With A Localized Source In A Ball

For decades, the study of the diffusion equations has been greatly improved, and a large number of literature has been devoted , especially, to the linear case. In 1984, Weissler first investigated the Dirichlet problem of the equationHe obtained in one dimensional case the single point blow-up phenomena. Then, Friedman-Mcleod and Fujita-Chen extended his results to N dimensional case. They showed that the blow-up point is only the origin, namely, S = {0}.Some other authors considered the Dirichlet problem of the equation with localized sourceIn 1992, Chadam et al. studied the Cauchy problem, the Neumann problem and the Dirichlet problem of the above equation with x*(t) (?) x*. They obtained the blow-up conditions for these problems. Moreover, they proved that total blow-up occurs whenever a solution blow-up. Souplet improved their results to the case for moving source x*(t) under the Dirichlet condition and obtained the precise profiles of total blow-up solutions.Fukuda-Suzuki studied Dirichlet problem of the equation with both local and nonlocal reaction termsThey made complete classification of total blow-up phenomena and single point blow-up phenomena according to p and q. So far as nonlinear diffusion equations (m > 2) are concerned, there have been a lot of results as well, on the existence, uniqueness and asymptotic behaviors. However, there are not so many results in the respect of the blow-up behavior. Thanks to the forerunners' achievement, the purpose of this work is to investigate the case when m > 2. Conditions for single point blow-up are obtained. These problems mainly come from non-Newtonian fluid mechanics(dilatant fluids have m > 2).Considerwhere . It is assumed throughout this paper·(A2) u0 is radially symmetric, u0(r) is non-increasing for r∈[0, R].In section2, some lemmas are delivered. First, from the assumption (A2), there holdsLemma 1. ur < 0 for all (x, t)∈B(R)×[0, T). The following Lemma2 reveals the effects of the localized reaction term on solution.Lemma 2. Suppose (A1) and (A2). Let u be the blow-up solution of problem (3)-(5) with blow-up time T. If , then single point blow-up occurs, namely, S = {0}. Many of the results on blow-up have identified the maximum existence time(or the blow-up time) , and the blow-up time T in this work has the following estimate.Lemma 3. Let m - 1 < p , D be a domain in B(R) with smoothboundary (?)D. For anyε> 0, there exists h0 such that T <ε, ifThe following Lemma4 and Lemma5 are crucial and essential to the main results.Lemma 4. Let andγ> 0. Assume (A1, (A2)and 0 < q < p(p > m - 1) or p = q > 2. PutChooseγ> 0 so small thatω(α,γ) (?) B(R1) \ B(R2), and define, for the solution u of problem (3)-(5)where . Let u be the solution of problem (3)-(5). Then there exists a constant M > 0 suchthat if inf u(x,t)≥M for someτ∈[0,T), the inequalityholds inω(α,γ)×(Ï„, T), for anyε∈(0,1), where and B = |u_r|^m-1.Lemma 5. Let satisfy 0 < R3 < R2 < R1 < |x*\. Denote R₀ = R. Assume p < q. Let hi(r)∈C²([0,∞))(i = 1,2,3) satisfy thatand Define, for l, k≥1, for x∈B(R),where r = |x|.Let u be a solution of (3)-(5) with the initial data u₀(x) = v_l,k(x). For any K > 1, there exists a constant l₀ = l(K)(independent of k) such that the following holds: For any l≥l₀, there exists M = M(K, l) > 0(independent of k)such that ifThenMoreover, if T' = T <∞, then the solution u blows up only at the origin. Due to the above lemmas, it goes to the main results:Theorem1. Let x*≠0. Assume (A1, (A2) hold. 0 < q < p(p> m - 1) or p = q > 2. Then the solution of (3)-(5) only blows up at the origin, namely, S = {0}.Theorem2. Assume (A1), (A2) and m - 1 < p < q. Let x*≠0. Then the solution of (3)-(5) blows up at the origin if u0(x) is sufficiently large.