Global Solvability Of Nonlinear Diffusion Equations With Forcing At The Boundary

Nonlinear diffusion equations have profound physical background as a kind of important parabolic partial differential equations. They are the mathematical formulation of the diffusion phenomenon existing generally in nature. Nonlinear diffusion equations involved the scientific research fields in many aspects including physics, chemistry and biotic colony dynamics etc. Among which, the most essential and important type is Newtonian filtration Equation:This equation has attracted numerous mathematicians' attention both in China and abroad as early as thirty years ago. They took up with the researches on theory and application of this class of equation, in including the existence, uniqueness, asymptotic property and blow-up etc. There have been a lot of corresponding literatures dealing with the results.Filtration is a kind of common phenomenon in nature, which indicates the movement of the fluid in porous medium. The research on it plays an important role in exploitation of ground water resouces and the discovery of petroleum or gas, especially to the agriculture. At the same time, when we investigate the problem about the saline-alkali soil and melioration, the using fertilizer intelligently, the industrial waste water disposal and the protection of the ground water resource, which are involved the solute movement and the heat transfer, we must consider the dynamics of the solute in the filtration and the heat transportation.The research of filtration phenomenon originated from the famous experiment of H. Darcy in 1956. In the following decades, many mathematicial models were established, and researches on numerical computation and the theoretical qualitative analysis have been achieved a great deal progress.The focus of this paper is the global existence of nonnegative solutions of this Newtonian filtration equationin which, a,ε,m,n,p＞0, u₀∈L^∞((0, 1)) and u₀≥0. in addition, the following condition is satisfied:(H₁)g,f,h∈C(R),g′,f′,h′∈C (R\{0}), g(0)=f(0)= h(0)=0,g′(u)＞0 for u＞0, u₀∈L^∞((0, 1));(H₂)|f′(u)|=O(u^n-1), |h(u)|=O(u^q), |φ(u)|=O(u^p), u→∞;(H₃)There are m＞0 andα＞0 to make all the u＞0 satisfyg′(u)≥αu^m-1, 4α≥(r+m)², g(u)/g′(u)≤C_αu.This thesis will testify the global existence of solution to problem (A). That is, the solution u to the problem of (A) exists for all time t. The discussion on (A) originates from the porous medium model. Under the influence of a sufficiently "weak" nonlinear source term, it is by now well known that a degenerate diffusion equation is globally solvable. A similar result is known when the nonlinear source is present as a forcing term at the boundary. Such results are usually established via comparison with solutions of a related differential equation or some other form of the maximum principle. However, these techniques do not appear to apply in general situations when forcing occurs on only part of the boundary and convection is present. In this work, we obtain the global existence of solutions for such problems by deriving a differential inequality involving the L^(?)+1 norms of solutions. The result is similar to that described above for equations with a nonlinear source term and, furthermore, establishes the global existence of solutions even for "strong" forcings. The technique also applies to a more general class of problems involving nonlinear reaction, diffusion, and convection.The solution which are constructed by J. R. Anderson (Comm. Partial Differential Equations 16, No. 1(1991), 105-143) hereafter referred to as the limitsolutions of (A) are obtained via the pointwise limit of a sequence {u_k,j}_j=1^∞with k→0⁺. Each of the terms u_k,j is a classical solution of a regularized problem, and an estimate of‖u_k,j(·,t)‖_∞for some k＞0 which is finite for all t＞0 and independent of j is sufficient for establishing an a priori estimate of‖u^L(·,t)‖_∞for all t＞0, where u^L≡lim_k→0lim_j→0u_k,j is the limit solution. There is acontinuation result for u^L, in the sense that eitheror u^L can be continued to be a solution on (0, 1)×(0, T+δ), forδ＞0. The global existence of the limit solution can thus be concluded. The arguments used herein require more regularity than is currently known to be true for u^L, and thus, we first work with the approximating sequence {u_k,j} in the direction of obtaining the necessary estimate of‖u_k,j(·,t)‖_∞. The limit solution always turns out to be the maximal solution of (A). That is, if u is any solution of (A) with u^L(x,0)=u(x,0)=u₀(x), then u≤u^L on (0,1)×(0,T). Therefore, even in theabsence of a uniqueness result, if we know that solutions of (A) are continuable in the sense discussed above, then the global existence of all nonnegative solutions follows.In the direction of obtaining such a theory for problem (A), we prove that if any of the following are satisfied:(1)max{n,p,ε}＜m(2)n＜m=p=εand a≥0, b≥0 is small enough(3)p=q＜m=n andε≥0 is small enough(4)p=q=m=n and a, b,ε≥0 is small enough, then the limit solution of (A) exists on (0,1)×(0,∞) for any choice of u₀. Furthermore, the global existence of initially "small" solutions whenever n,p≥m will also follow from our work. In the remaining cases p＜m＜n or n＜m＜p, we show that initially "small" solutions are again global provided the parameters a or∈, respectively, are appropriately "small".This paper mainly discusses the global existence of nonnegative solutions of the parabolic equation with forcing at the boundary. The basic idea is the limit solution method: we get the estimation of the L^(?)+1 norm of the solution. Since the problems what we consider involving more general reaction, diffusion, convective and boundary fluxes, there are many difficulties in obtaining the estimation of solution. When r＞1/2, m≥1, we can get the estimation of L^∞norm of u_k by adopting the monotonicity and maximum principle and come to the conclusion that the non-negative global solution is existent. However, this method can not apply to the case of m＜1. Furthermore, it is impossible to get the estimation of the L^∞norm through limit r→∞. Therefore, we verify a lemma that provides us with the essential estimation and thus to get the estimation of L^∞norm of the solution.DEFINITION 1. A function u(x, t) defined on [0,1]×[0,T] is called a solution of (A) on (0,1) x (0,T)if u satisfies:(â…°) u∈L^∞((0, 1) x (0, T)),(â…±) u=0 on {0} x (0, T) and u = u_o on [0,1] x {0},(â…²) For every t∈[0, T] and every functionζsuch thatζ(0, t) = 0,ζ≥0, andζ_t,ζ_x,ζ_xx∈L²((0, 1) x (0, T)).THEOREM 1. (Local Existence and Continuation). There exists a T＞0, which depends on‖u₀‖_∞, such that (A_k) possesses a unique solution, u_k, on (0, 1) x (0, T) for each k∈(0, 1). Furthermore, u≡lim_k→0+u_k is a solution of (A) on (0,1) x (0, T). If T≡T(u₀)＞0 is defined to be the largest value of t such that u is a solution of (A) on (0, 1) x (0, s) for all s＜t, thenLEMMA 1. Let u denote the solution of (A) on (0, 1) x (0, T). If F (z)＜0 for Z₁＜z＜z₂ and iffor some r＞m satisfying r + 1≥max{2n, 2p, 2q} ,then‖u‖_r+1^r+m＜z₂ for all t∈(0,T).LEMMA 2. Let u denote the solution of (A_k) on (0, 1) x (0, T), where k＞0, and assume . There exists a constant C, which depends on k, such thatTHNORNM 2. (Global Existence of the Limit Solution of (A)). Let r＞m be chosen to satisfy r+ 1≥maz{4n, 2p}. Under any of the following assumptions the limit solugion of (A) having initial daga u₀ exist on (0, 1) x (0,∞).(â…°)Aay of the conditions (1)-(4) are saisfied, and u₀∈L^∞(0,1) with u₀≥0. (â…±)p, n≥m, with a≥0 assumed to be sufficiently small in case p = m and∈≥0 sufficiently small in case n = m, andfor some constant C＞0.(â…²)p＜m＜n, a≤(?)_∈^-(m-p)/(n-m), andfor some constant C＞0.(â…³)n＜m＜p,∈≤(?)_a^-(r+m)/(p-m), andfor positive constants (?) and C.THEOREM 3. (Eventual Mass of Global Limit Solutions). Let u denote a global limit solution from Theorem 2.2 or 2.3, and let r＞m satisfy r + 1≥max{4n, 2p}. There exists a constant M＞0 such that lim_{t→∞}sup‖u‖_r+1^r+m≤M. Moreover, in cases of Theorem 2.2(â…±)-(â…³) or 2.3(â…±) above, we have M = 0.