Nonlinear Diffusion System With Evolution P-Laplacian In The Presence Of Fast Reaction

In this paper, we consider the following initial boundary value problemwhere Qt :=Ω× (0,T], Ω is a bounded domain in Rⁿ, T, u0, v₀ are positive constants, and F : R×R —> R satisfies the following assumptions:(H₂) F is nondecreasing with respect to the two variables;(H₃) F(0,s) = F(s,0) = 0, s G [0, v0], s € [O,u_o], This problem is a mathematical model describing the interaction between virus and antibody, where u denotes the concentrationof anti-body, v denotes the concentration of virus, and F(u, v) denotes the reaction-control function. Broad attentions have been focused on this kind of models in recent years. Among them, D. Hilhorst and others studied the reaction-diffusion system of equations to virus and antibody in one dimensional space and multidimensional space in a sequence of papers, where the diffusion term of the antibody equation were Aw and Δφ(u) respectively. They proved the existence, uniqueness, and the convergence of the solutions as k —>∞. Ke Changhai and Gao Wenjie studied the problem with diffusion term in the equation to virus in their recent work. They studied the existence, uniqueness, and also the convergence of the solutions as k —> ∞.In this paper, we will replace the orginal equation to the antibody in Ke's paper by the well-known diffusion equation- the evolution p—Laplace equation. We will prove the existence, uniqueness, and the convergenceof the solutions as k —> ∞ . Our method is different from D. Hilhorst's and Ke's method since there is not necessarily a classical solution for the equation we considered. Therefore we have to consider the weak solutions throughout this paper.The paper is organized as the following:(I) The existence of the solutions to the systemsWe first get some a prior estimates to the solution of the systems. To study the properties of the solution w, we introduce thefollowing problemut = dw(\Vu\p-2Vu) - kF(u, u>), (x, t) G QT, (5)u = u0, (x,t) edQx{Q,T], (6)u(x, 0) = 0, x e Q, (7)here 0 < uj < vq.We haveLemma 1.1 Assume uj G L°°(Qt),0 < oj < vo, then there exists a unique solutionu G C(QT\(dQ) x {0}) to (5)-(7) satisfying(a) 0 < u <(6) // \Vu\pdxdt 0. Similarly, for the virus, we introduce the following problem vt = Av- kF(uj, v), (x, t) G QT, (8)v{x, 0) = v0, xe Q, (10)with 0 < uj < uq.We haveLemma 1?.2 Assume uj G L°°(Qt),0 < w < wq, then there exists a unique solutionv G C(QT\(dQ.)x{0}) to (8)-(10) satisfying(a) 0 0.The existence of the solutions depends on these lemmas.With the use of lemma 1.1 and lemma 1.2, we may construct two sequence of the solutions {um}, {vm}, and prove that these sequences are monotone. Then by using the Bootstrap method, we prove the convergence of these sequence and hence obtain the existence of the solutions.Theorem 1.1 There exists a pair of weak solutions (u,v) to (l)-(4) satisfying(a) 0 < u < wo, 0 < v < vo, (x, t) G Qt;(b) ut>0, vt<0 a.e.T{x,t)eQT-Then we prove the uniqueness of the solution.Theorem 1.2 There exist at most one solution to the system(II). The limit case to fast reactionSince the system is introduced to the reaction and diffusion between the anti-body and virus, we should consider the limit to fast reaction, that is the limit of the solutions when k —> oo. First we prove some a prior estimates.Lemma 1.3 There exists a constant C independent ofk, s.t. (a) ff \Vvk\2dxdt oo(a) ukn converges strongly to U in L2(Qt);(b) uk — uq converges weakly to U — uq in i/(0, T; Wq'p(Q));(c) vkn converges strongly to V in L2(Qt);(d) vkn converges weakly to V in L2(0,T;Hq(Q)).By studying the limit of the solutions as k —> oo, we may explain some properties in the process of the fast reaction. The results show that as k goes to infinity, if the concentration of the anti-body remains positive, the concentration of the virus must go...