GLOBAL EXISTENCE AND UNIQUENESS OF CLASSICAL SOLUTIONS FOR A GENERALIZED QUASILINEAR PARABOLIC EQUATION WITH APPLICATION TO A GLIOBLASTOMA GROWTH MODEL

. This paper studies the global existence and uniqueness of classical solutions for a generalized quasilinear parabolic equation with appropriate ini- tial and mixed boundary conditions. Under some practicable regularity criteria on diﬀusion item and nonlinearity, we establish the local existence and unique- ness of classical solutions based on a contraction mapping. This local solution can be continued for all positive time by employing the methods of energy estimates, L p -theory, and Schauder estimate of linear parabolic equations. A straightforward application of global existence result of classical solutions to a density-dependent diﬀusion model of in vitro glioblastoma growth is also presented.


(Communicated by Haiyan Wang)
Abstract. This paper studies the global existence and uniqueness of classical solutions for a generalized quasilinear parabolic equation with appropriate initial and mixed boundary conditions. Under some practicable regularity criteria on diffusion item and nonlinearity, we establish the local existence and uniqueness of classical solutions based on a contraction mapping. This local solution can be continued for all positive time by employing the methods of energy estimates, L p -theory, and Schauder estimate of linear parabolic equations. A straightforward application of global existence result of classical solutions to a density-dependent diffusion model of in vitro glioblastoma growth is also presented.
1. Introduction. In this paper, we consider the existence and uniqueness of classical solutions for a generalized quasilinear parabolic equation of the form with initial and mixed boundary conditions u(x, 0) = u 0 (x) in Ω,

WEN, FAN, ASIRI, ALZAHRANI, EL-DESSOKY AND KUANG
where Ω ⊂ R n is a bounded open domain with the boundary ∂Ω being C 2+α (0 < α < 1), ν is the unit outward normal vector on ∂Ω and ∇u = (u x1 , · · · , u xn ). Throughout this paper, we assume that A : Ω × R + × R × R n → R n and F : Ω × R + × R × R n → R, and A = A(x 1 , · · · , x n , t, u, p 1 , · · · , p n ) satisfies the strongly uniformly parabolic condition ∂A j ∂p i (x, t, u, p)ξ i ξ j ≥ µ 1 |ξ| 2 > 0 for all ξ = (ξ 1 , · · · , ξ n ) ∈ R n . In the last few decades, many quasilinear parabolic equations, equipped with appropriate initial or boundary conditions, have been widely investigated to explain and predict the real-world phenomena in areas such as chemistry, physics, biology, ecology. For example, some of these mathematical models can be applied in depicting various biological processes, such as bacterial growth process, development and growth of tumors, immune response of the body, see [5,14] and [18].
Mathematically, quasilinear parabolic systems have been extensively studied by the methods in nonlinear analysis and theory of PDEs. Besides the existence of time-dependent solutions which have been discussed in [2,3,4,11,13,15,17,20] and [21], research efforts have also focused on spatial or spatio-temporal patterns in [9,12] and [16]. While [7] and [8] investigated traveling waves or other types of entire solutions, and in addition to numerical simulations by finite element methods with small diffusion found in [6] and [10].
In [11], three types of generalized quasilinear equations, i.e., (1) and for the first boundary value problem, and for other boundary value problem are considered. Based on Leray-Schauder principle, the local solvability of classical solutions to these systems are established provided that |u| and |u x | are both bounded on Ω × (0, T ), F and the derivatives of A (or the second derivative of a ij ) satisfy some regularity or growth restrictions. In a series of papers on dynamic theory for quasilinear parabolic equations ( [2], [3], [4]), Amann discussed the local and global existence of classical solution for a general second order quasilinear parabolic systems. According to Amann's results, equation with initial value of W s,p class and homogeneous Neumann boundary condition has a unique solution if s, p are chosen properly, and for some constant κ ≥ 1 and an increasing function c. Moreover, the regularity assumption for F and condition (7) have only been used in the proof of the existence and regularity of u. If we are already in possession of a classical solution u, it suffices to assume for the global existence of u. Based on these results and comparison theorem, [22] gave some conditions on the nonlinear part F for the global-in-time existence of classical solution to equation (5) under homogeneous Neumann boundary condition. However, for some concrete forms of equations (1) and (3)- (5), diffusion and nonlinearity functions may have low regularity than those in (6) or can not meet so many assumptions in §5.6, §5.7 and §6.4 of [11]. Consequently, the results available in the literature are not readily applicable to these cases. Therefore, there are practical needs for additional studies for establishing both local and global existence of classical solutions to the generalized quasilinear parabolic equations under weaker regularity or fewer growth restrictions.
In this paper, we investigate the global existence and uniqueness of classical solution to problem (1)- (2). In Section 2, under some continuity conditions on diffusion and growth terms, the local existence and uniqueness of classical solution are established by using the contraction mapping theory. Next, we perform some priori estimates, and then show that the local solution can be extended to entire time interval in Section 3. In Section 4, we apply the main results to two specific examples, including a recently data-validated glioblastoma growth model.

2.
Local existence and uniqueness. In this section, we explore the local existence and uniqueness of classical solution to problem (1)- (2). The approach is based on the theory of fixed point.
(H1) A(x, t, u, p) is differentiable in the variables x, u and p. and F are locally C α,α/2 continuous with respect to (x, t) and locally Lipschitz continuous with respect to (u, p), uniformly with respect to the other variables.
(H3) σ is twice differentiable in the variable x and differentiable in the variable t, each of these derivatives is locally C α,α/2 continuous with respect to (x, t) while locally Lipschitz continuous with respect to u. Further, σ satisfies the compatibility condition of 0-order: Proof. Define For any u ∈ X B , define a mapping G : u →ũ, whereũ satisfies From u ∈ X B , (H1), (H3) and the definition of Hölder spaces, it is easy to verify that b ij , b j , f 1 ∈ C α,α/2 (Q T ), σ 1 ∈ C 2+α,1+α/2 (Γ T ). By Theorem 5.3 (pp.320-321) in [11], (9) possesses a unique solutionũ ∈ C 2+α,1+α/2 (Q T ) satisfying By the norm definition in (8) and differential mean value theorem, one has which implies that for sufficiently small T relations hold true. Hence,ũ ∈ X B , i.e., G maps X B into itself. Next, we show that G is contractive. Let u, v ∈ X B ,ũ = Gu,ṽ = Gv. We only need to verify Denotew =ũ −ṽ. Then,w satisfies where b ij and b j are given in (10), By Cauchy inequality, (H1), (H3) and (11), we have where L is the maximum of Lipschitz constants. Notice that, b ij (x, t) is bounded continuous in Q T . By Theorem 9.1 (pp.341-342) in [11], one has for any q > 3. (15) Lemma 3.3 (p.80) in [11] implies that and then Take T = T 0 small enough such that then inequality (13) holds. By the classical contraction mapping theorem, G has a unique fixed point u ∈ X B , which is the unique local solution of (1)- (2). Taking T 0 as initial time and u(·, T 0 ) as initial value, one can continue the solution to a larger time interval. The procedure may be repeated indefinitely leading to the construction of a maximally defined solution u ∈ C 2+α,1+α/2 (Ω × [0, T )) for some T > 0.
3. Global existence for the case A(x, t, u, ∇u) = a(x, t, u)∇u. In this section, we investigate the global existence of classical solution to system (1)-(2) by extending the existence interval of the local solution to [0, +∞). Let a : Ω × R + × R → R. We only consider the special case of A(x, t, u, ∇u) = a(x, t, u)∇u and σ ≡ 0, i.e., with initial and boundary conditions According to the discussion in Section 2, we assume that

413
(H1) * a(x, t, u) is differentiable in the variables x, t and u. a, ∂a ∂xi , ∂a ∂u and F are locally C α,α/2 continuous with respect to (x, t) and locally Lipschitz continuous with respect to u, uniformly with respect to the other variables. Further, there exist positive constantsμ 1 ,μ 2 such that In order to establish the non-negativity and L p -estimate of solutions for system (17)-(18), we further assume that the nonlinear part satisfies Now, we investigate the non-negativity of solution to problem (17)- (18). Note that, if u is a classical solution to (17)-(18), then we have u ∈ C 2,1 (Q T ) ∩ C(Q T ). Assume that u(x, t) is a classical solution to (17)- (18). Then, u also satisfies the following differential inequality Considering the assumptions (H1) * , one can easily verify that a ij , a j are all continuous. By the maximum principle for linear parabolic equation, one has u ≥ 0 in Q T .
Then, similar to the proof of Theorem 2.1, we get a local existence conclusion for problem (17)- (18). In the following discussion, denote by C(T ), C i (T )(i = 1, 2, · · · ) the constants depending not only on the parameters and initial value in (17)-(18) but also on time span T , and by C, C i (i = 1, 2, · · · ) the constants only depending on the parameters and initial value in (17)-(18).

A priori estimates.
To continue the local solution established in Theorem 2.1, we need to perform some a priori estimates for the unknown function and its derivative.
In order to achieve higher regularity of |∇u|, we introduce the following lemma.
Thus, Du ∈ L 4 (Q T ). A standard L p estimate for linear parabolic equation shows that u ∈ W 2,1 4 (Q T ). Similar to estimate (27), one has And then, Now, we are at the right position to establish the boundedness of u. Since the coefficients a 2 ij , a 2 j , f * of equation (28) are in L 3 (Q T ), all conditions of Theorem 7.1 (pp.181-182) [11] are fulfilled for n ≤ 3. Therefore, there must be a positive constant C 27 (T ) such that u L ∞ (Q T ) ≤ C 27 (T ).
By Remark 2 and the proof of Lemma 3.4, we can easily prove the following result similar to Lemma 3.5. Lemma 3.6. Let u be a classical solution of (17)- (18) and n ≤ 3. Then, for any positive integer k, Du ∈ L 2 k+1 (Q T ) and Du L 2 k+1 (Q T ) C(T ).
Proof. We prove it by contradiction arguments. Assume that [0, T * ) is the maximum existence interval of the solution to (17)- (18). For any ∈ (0, T * ), take u(x, T * − ) as the new initial value. By Theorem 2.1, one can extend the solution to Q (T * − )+T1 for some T 1 > 0, here T 1 only depends on the upper bound of u(x, T * − ) C 2+α (Ω) . By the a priori estimate in Lemma 3.7, T 1 only depends on T * but does not depend on . Hence, we can choose an appropriate satisfying < max{T * , T 1 } and then (T * − ) + T 1 > T * , which contradicts to the definition of T * . The proof is complete.

Remark 3.
It is easy to see that a ∈ C 2 (Ω × R + × R, R) and F ∈ C 1 (Ω × R + × R × R n , R) are sufficient to ensure (H1) * holds true. Example 1. In order to model the glioblastoma tumor growth, [19] proposed a density-dependent convective-reaction-diffusion equation, whose one-dimensional Cartesian coordinate version reads where the behavior of both proliferation and migration processes are incorporated. In (30), diffusion is large for areas where the amount of cells are small (the migrating tumor cells), but diffusion is small where the cell density is large (the proliferating tumor cells). There are many functions that could serve as the diffusion function D(u), for example, D(u) = D 1 − D 2 u n a n + u n , here D 1 , D 2 , a, n are all positive constants, n > 1, and D 2 ≤ D 1 to avoid negative diffusion.
Although the dynamics of (30) are well explored, the existence and uniqueness of solutions are not investigated since the criteria existing in the literature can not be applied. We claim that the existence and uniqueness of (30) satisfying the boundary condition (18) falls into the framework of this study.
This shows that D(u) and D (u) have bounded derivatives on [0, ∞). Clearly, D(u), D (u) are both Lipschitz continuous on [0, ∞), and F (u, p) = u(1 − u) − γp is locally Lipschitz continuous with respect to u or p.

5.
Discussion. This paper is devoted to the study of the global-in-time solutions for a generalized quasi-linear parabolic equation with applications in biology and medicine. Under some practical regularity and structure conditions on diffusion term and nonlinearity, we establish the local and global existence and uniqueness of classical solutions for problem (17)- (18). The main results are Theorems 3.1, 3.8 and 3.9, which show that the unique solution of problem (17)-(18) and its derivatives (u, u xi , u xixj and u t ) are all continuous in Q T . One of the main difficulties for global existence is to perform the L p -estimate of Du. Comparing to the existing results on the global solutions of quasi-linear parabolic equations, our conditions on diffusion item and nonlinearity function represented by ((H1) * , (H4) − (H6)) are easier to verify and need weaker regularity.
However, we only investigate the existence of classical solutions for a singleequation system in the present paper. In real-world applications, quasi-linear parabolic systems consisting of two or more equations are more common and significant. Thus, we are also interested in the existence of classical solutions for quasi-linear parabolic systems of equations. The method of classical fixed point theory is usually effective for studying the question of local existence of solutions to systems of equations. But the L p -estimate techniques for the unknowns and their gradients are more difficult due to the coupled diffusion and nonlinear terms. To overcome this difficulty, we may need some novel Sobolev embedding results and new interpolation inequalities. We encourage future efforts along these directions.