Renormalized Solutions of a Nonlinear Parabolic Equation with Double Degeneracy ∗

In this paper, we consider the initial-boundary value problem of a nonlinear parabolic equation with double degeneracy, and establish the existence and uniqueness theorems of renormalized solutions which are stronger than BV solutions.


Introduction
This paper is concerned with the following initial-boundary value problem B(u(0, t)) = B(u(1, t)) = 0, t ∈ (0, T ), ( u(x, 0) = u 0 (x), x ∈ (0, 1), (1.3) where f (s) is an appropriately smooth function and with p ≥ 2 and b(s) ≥ 0 appropriately smooth.The equation (1.1) presents two kinds of degeneracy, since it is degenerate not only at points where b(u) = 0 but also at points where Using the method depending on the properties of convex functions, Kalashnikov [10] established the existence of continuous solutions of the Cauchy problem of the equation (1.1) with f ≡ 0 under some convexity assumption on A(s) and B(s).Under such assumption, the equation degenerates only at the zero value of the solutions or their spacial derivatives.The more interesting case is that the equation may present strong degeneracy, namely, the set E = {s ∈ R : b(s) = 0} may have interior points.Generally, the equation may have no classical solutions and even continuous solutions for this case and it is necessary to formulate some suitable weak solutions.
For the semilinear case of the equation (1.1) with p = 2, it is Vol'pert and Hudjaev [12] who first introduced BV solutions of the Cauchy problem and proved the existence theorem.Later, Wu and Wang [13] considered the initial-boundary value problem.For the quasilinear case with p > 2, the existence and uniqueness of BV solutions of the problem (1.1)-(1.3)under some natural conditions have been studied by Yin [15], where the BV solutions are defined in the following sense 2) and (1.3) in the sense that where u r (•, t) and u l (•, t) denote the right and left approximate limits of u(•, t) respectively and ū denotes the symmetric mean value of u.
(ii) for any k ∈ R and any nonnegative functions In this paper we discuss the renormalized solutions of the problem (1.1)-(1.3).Such solutions were first introduced by Di Perna and Lions [8] in 1980's, where the authors studied the existence of solutions of Boltzmann equations.From then on, there have been many results on renormalized solutions of various problems, see [1,2,3,4,5,6,7,11].It is shown that such solutions play an important role in prescribing nonsmooth solutions and noncontinuous solutions.The renormalized solutions of the problem (1.1)-(1.3)considered in this paper are defined as follows 3) in the sense of (1.4) and (1.5); (ii) for any k ∈ R, any η > 0 and any nonnegative functions the following integral inequality holds where Such solutions are a natural extension of classical solutions, which will be shown at the beginning of the next section.Comparing the two definitions of weak solutions, there are two additional terms in (1.7), i.e. the forth and fifth ones.Moreover, (1.6) follows by letting η → 0 + in (1.7) since the forth term of (1.7) is nonnegative and the limit of the fifth term is zero.Therefore, renormalized solutions imply more information than BV solutions and thus it is stronger.
Since renormalized solutions are stronger than BV solutions, the uniqueness of renormalized solutions of the problem (1.1)-(1.3)may be deduced directly from the uniqueness of BV solutions (see [15]).Hence The paper is arranged as follows.The preliminaries are done in §2.We first prove that the classical solution is also a renormalized solution, which shows that the latter is a natural extension of the former.Then we formulate the regularized problem and do some a priori estimates and establish some convergence.Two technical lemmas are introduced at the end of this section.The main result of this paper (Theorem 1.2) is proved in §3 subsequently.

Preliminaries
The renormalized solution is a natural extension of the classical solution.In fact, we have Then for any k ∈ R, any η > 0 and any nonnegative functions 3) with (2.1), then u is also a renormalized solution of the problem (1.1)-(1.3)and the inequality (1.7) can be rewritten as the equality.
On the one hand, multiply (1.1) with H η (u − k)ϕ 1 and then integrate over Q T to get From the definition of F η (s, k), (2.1) and the Newton-Leibniz formula, Then, by using the formula of integrating by parts in (2.4) and from (2.5), we get On the other hand, the equation (1.1) leads to Multiplying this equation with ϕ 2 and then integrating over Q T , we get that by the formula of integrating by parts and (2.1) Since the equation (1.1) presents double degeneracy, we regularize the equation to get the existence of renormalized solutions by doing a prior estimates and passing a limit process.We firstly approximate the given initial data u 0 .For any 0 < ε < 1, choose u 0,ε ∈ C ∞ 0 (0, 1) satisfying where C > 0 is independent of ε, and EJQTDE, 2006 No. 5, p. 5 Consider the regularized problem u ε (0, t) = u ε (1, t) = 0, t ∈ (0, T ), (2.9) x ∈ (0, 1). (2.10) By virtue of the standard theory for uniformly parabolic equations, there exists a unique classical solution u ε ∈ C 2 (Q T ) of the above problem.To pass the limit process to the problem (1.1)-(1.3),we need do a priori estimates on u ε .On the one hand, the maximum principle gives sup Here and hereafter, we denote by C positive constants independent of ε and may be different in different formulae.On the other hand, the following BV estimates and C 1,1/2 estimates have been proved by Yin [15].
Lemma 2.1 The solutions u ε satisfy Lemma 2.2 For the function we have Form the estimate (2.11), Lemma 2.1 and Lemma 2.2, there exist a subsequence of ε ∈ (0, 1), denoted by itself for convenience, and a function u and (1.4) and (1.5), and a function µ ∈ L ∞ (Q T ), such that a.e. in Q T , (2.12) as ε → 0 + , see more details in Yin [15].
To complete the limit process, we also need the following convergence.
EJQTDE, 2006 No. 5, p. 6 Lemma 2.3 For the solution u ε of the regularized problem (2.8)-(2.10)and the above limit function u, we have Proof.We first prove that (2.17) For convenience, we rewrite Multiplying (2.8) with (B ε (u ε )−B(u)) and then integrating over Q T , we get that by the formula of integration by parts . On the other hand, by (2.14) and B(u where Then, from the definition of a * ε , (2.21) and (2.20), lim This, together with (2.15) yields (2.17), namely which deduces (2.16).The proof is complete.
In order to reach Theorem 1.2, we need the following two technical lemmas, which may be found in [9], [14].
Assume that A(s), B(s) are continuous functions, and A(s) is nondecreasing.If for any α ∈ A(R), B(A −1 (α)) contains only a single point, and Lemma 2.5 Let Ω ⊂ R n be a bounded domain and 1 < q < ∞.Assume {f m } is a sequence in L q (Ω) and f ∈ L q (Ω) with f m f, weakly in L q (Ω) as m → ∞.

Proof of the Main Result
In this section, we will complete the proof of Theorem 1.2 based on the estimates and convergence established in §2.
Proof of Theorem 1.2.For any k ∈ R, any η > 0 and any nonnegative functions By the definitions of H η (s, k) and F η (s, k), and by using Lemma 2.4 with (2.12), we get Combining this with (2.16) and (2.12), we have lim which will be proved below.Multiplying (2.8) by u ε and then integrating over Q T , we derive that by the formula of integration by parts These two estimates and (3.5) yield From the definitions of A ε and B ε , the above inequality leads to . This is just (3.4).The proof of Theorem 1.2 is complete.