Renormalized Solutions of a Nonlinear Parabolic Equation with Double Degeneracy

Electronic Journal of Qualitative Theory of Differential Equations 26, No. 5, -2; http://www.math.u-szeged.hu/ejqtde/ Renormalized Solutions of a Nonlinear Parabolic Equation with Double Degeneracy Zejia Wang,2, Yinghua Li 2 and Chunpeng Wang 2 Department of Mathematics, Northeast Normal University, Changchun 324, P. R. China 2 Department of Mathematics, Jilin University, Changchun 32, P. R. China bstract 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. Keywords: Renormalized solutions, double degeneracy. 2 MR S. C.: 35K6, 35K65 Introduction This paper is concerned with the following initial-boundary value problem u t + f(u = [ B(u, (x,t (, (,T, (. B(u(,t = B(u(,t =, t (,T, (.2 u(x, = u (x, x (,, (.3 where f(s is an appropriately smooth function and (s = s p 2 s, B(s = s b(σdσ, s R with p 2 and b(s appropriately smooth. The equation (. presents two inds of degeneracy, since it is degenerate not only at points where b(u = but also at points where B(u = if p > 2. Using the method depending on the properties of convex functions, Kalashniov [ established the existence of continuous solutions of the Cauchy problem of the equation (. with f under some convexity assumption on (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 = } 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 wea solutions. Supported by the NNSF of China and Graduate Innovation Lab of Jilin University. Corresponding author. Tel.:+86 43 5675. EJQTDE, 26 No. 5, p.

For the semilinear case of the equation (. with p = 2, it is Vol pert and Hudjaev [2 who first introduced BV solutions of the Cauchy problem and proved the existence theorem. Later, Wu and Wang [3 considered the initial-boundary value problem. For the quasilinear case with p > 2, the existence and uniqueness of BV solutions of the problem (. (.3 under some natural conditions have been studied by Yin [5, where the BV solutions are defined in the following sense Definition. function u L ( BV ( is said to be a BV solution of the problem (. (.3, if (i u satisfies (.2 and (.3 in the sense that B(u r (,t = B(u l (,t =, a.e. t (,T, (.4 lim ū(x,t = u (x, a.e. x (,, (.5 t + 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 R and any nonnegative functions ϕ i C ( (i =,2 with suppϕ i [, (,T and ϕ (,t = ϕ 2 (,t, ϕ (,t = ϕ 2 (,t, < t < T, the following integral inequality holds [ sgn(u (u ϕ ( t + (f(u f( ϕ B(u ϕ [ + sgn (u ϕ ( 2 t + (f(u f( ϕ 2 B(u ϕ2 dxdt dxdt. (.6 In this paper we discuss the renormalized solutions of the problem (. (.3. Such solutions were first introduced by Di Perna and Lions [8 in 98 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 [, 2, 3, 4, 5, 6, 7,. It is shown that such solutions play an important role in prescribing nonsmooth solutions and noncontinuous solutions. The renormalized solutions of the problem (. (.3 considered in this paper are defined as follows Definition.2 function u L ( BV ( is said to be a renormalized solution of the problem (. (.3, if (i u satisfies (.2 and (.3 in the sense of (.4 and (.5; (ii for any R, any η > and any nonnegative functions ϕ i C ( (i =,2 with suppϕ i [, (,T and ϕ (,t = ϕ 2 (,t, ϕ (,t = ϕ 2 (,t, < t < T, the following integral inequality holds H η (u, ϕ t dxdt + F η (u, ϕ dxdt EJQTDE, 26 No. 5, p. 2

B(u H η (u ϕ dxdt + H η(τ (f(τ f(dτ + H η ( B(u H η (ϕ (,t ϕ (,tdt [ (u ϕ ( 2 t + (f(u f( ϕ 2 B(u ϕ2 (u u ϕ dxdt dxdt, (.7 where H η (s, = H η (s = s s s 2 + η, H η (s = η (s 2, s R, + η 3/2 H η (τ dτ, F η (s, = s H η (τ f (τdτ, s, R. 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 wea solutions, there are two additional terms in (.7, i.e. the forth and fifth ones. Moreover, (.6 follows by letting η + in (.7 since the forth term of (.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 (. (.3 may be deduced directly from the uniqueness of BV solutions (see [5. Hence Theorem. There exists at most one renormalized solution for the initial-boundary value problem (. (.3. nd we will prove the existence of renormalized solutions of the problem (. (.3 in this paper, namely Theorem.2 ssume u BV ([, with u ( = u ( =. Then the initial-boundary value problem (. (.3 admits one and only one renormalized solution. 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.2 is proved in 3 subsequently. 2 Preliminaries The renormalized solution is a natural extension of the classical solution. In fact, we have Proposition 2. Let u C 2 ( C( be a solution of the equation (. with u(,t = u(,t =, t (,T. (2. EJQTDE, 26 No. 5, p. 3

Then for any R, any η > and any nonnegative functions ϕ i C ( (i =,2 with suppϕ i [, (,T and ϕ (,t = ϕ 2 (,t, ϕ (,t = ϕ 2 (,t, < t < T, H η (u, ϕ t dxdt + F η (u, ϕ dxdt B(u H η (u ϕ dxdt + H η(τ (f(τ f(dτ + H η ( B(u H η (ϕ (,t ϕ (,tdt [ (u ϕ ( 2 t + (f(u f( ϕ 2 B(u ϕ2 (u u ϕ dxdt dxdt =. (2.2 Therefore, if u C 2 ( C( is a solution of the problem (. (.3 with (2., then u is also a renormalized solution of the problem (. (.3 and the inequality (.7 can be rewritten as the equality. Proof. Let R, η > and ϕ i C ( (i =,2 be nonnegative functions with suppϕ i [, (,T and ϕ (,t = ϕ 2 (,t, ϕ (,t = ϕ 2 (,t, < t < T. (2.3 On the one hand, multiply (. with H η (u ϕ and then integrate over to get ϕ t H η(u,dxdt + ϕ F η(u,dxdt = B(u H η (u ϕ dxdt. (2.4 From the definition of F η (s,, (2. and the Newton-Leibniz formula, = = ϕ F η (u, x= dt x= ( u(,t H η (τ f (τdτϕ (,t H η (τ (f(τ f( dτ =H η ( (f( f( u(,t (ϕ (,t ϕ (,tdt (ϕ (,t ϕ (,tdt H η (τ f (τdτϕ (,t H η (τ (f(τ f(dτ (ϕ (,t ϕ (,tdt. (2.5 Then, by using the formula of integrating by parts in (2.4 and from (2.5, we get H η (u, ϕ t dxdt + F η (u, ϕ dxdt dt EJQTDE, 26 No. 5, p. 4

= + H η ((f( f( + H η(τ (f(τ f(dτ ϕ (,t ϕ (,tdt (ϕ (,t ϕ (,tdt [ H η ( B(u(,t ϕ (,t B(u(,t ( + B(u H η (u ϕ + H η(u u ϕ On the other hand, the equation (. leads to ϕ (,t [ (u + (f(u f( = t B(u, (x,t. dt dxdt. (2.6 Multiplying this equation with ϕ 2 and then integrating over, we get that by the formula of integrating by parts and (2. (u ϕ 2 t dxdt + (f( f( [ = + B(u(,t B(u ϕ2 (f(u f( ϕ 2 dxdt (ϕ 2 (,t ϕ 2 (,tdt ϕ 2 (,t B(u(,t ϕ 2 (,t dt dxdt. (2.7 Multiplying (2.7 with H η ( and then adding what obtained to (2.6, we get (2.2 owing to (2.3. The proof is complete. Remar The condition (2. is not egregious. In fact, (2. and (.2 are identical if B(s is strictly increasing, i.e. the set E = {s R : b(s = } has no interior point. Otherwise, if the set E has interior points, the equation (. is strong degenerate and the equation may have no classical solutions in general, as mentioned in the introduction. Since the equation (. 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. For any < ε <, choose u,ε C (, satisfying u,ε dx C, where C > is independent of ε, and dx B ε(u,ε C, u,ε (x u (x, uniformly in (, as ε +, (s = (s + εs, B ε (s = B(s + εs, s R. EJQTDE, 26 No. 5, p. 5

Consider the regularized problem u ε t + f(u ε = [ B ε(u ε, (x,t, (2.8 u ε (,t = u ε (,t =, t (,T, (2.9 u ε (x, = u,ε (x, x (,. (2. By virtue of the standard theory for uniformly parabolic equations, there exists a unique classical solution u ε C 2 ( of the above problem. To pass the limit process to the problem (. (.3, we need do a priori estimates on u ε. On the one hand, the maximum principle gives sup u ε C. (2. 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,/2 estimates have been proved by Yin [5. Lemma 2. The solutions u ε satisfy Lemma 2.2 For the function sup u ε dx + sup u ε dx C, <t<t <t<t t sup dx <t<t B ε(u ε (x,t C, sup B ε(u ε C. ω ε (x,t = B ε (u ε (x,t, (x,t, we have ω ε (x,t ω ε (x 2,t C x x 2, x,x 2 (,, t (,T, ω ε (x,t ω ε (x,t 2 C t t 2 /2, x (,, t,t 2 (,T. Form the estimate (2., Lemma 2. and Lemma 2.2, there exist a subsequence of ε (,, denoted by itself for convenience, and a function u L ( BV ( with B(u C,/2 ( and (.4 and (.5, and a function µ L (, such that u ε (x,t u(x,t, a.e. in, (2.2 B ε (u ε (x,t B(u(x,t, uniformly in, (2.3 B ε(u ε B(u, wealy in L (, (2.4 B ε(u ε µ, wealy in L (, (2.5 as ε +, see more details in Yin [5. To complete the limit process, we also need the following convergence. EJQTDE, 26 No. 5, p. 6

Lemma 2.3 For the solution u ε of the regularized problem (2.8 (2. and the above limit function u, we have B ε(u ε (x,t B(u(x,t, a.e. in as ε +. (2.6 Proof. We first prove that For convenience, we rewrite (s = s µ = B(u, a.e. in. (2.7 a(σdσ, s R, a(σ = p σ p, σ R. Multiplying (2.8 with (B ε (u ε B(u and then integrating over, we get that by the formula of integration by parts u ( ε ( B ε (u ε B(u dxdt + t f(u ε B ε (u ε B(u dxdt = B ε(u ε B ε(u ε B(u dxdt, which yields lim ε + B ε(u ε B ε(u ε B(u dxdt = (2.8 from (2.3. On the other hand, by (2.4 and B(u C,/2 (, lim ε + B(u B ε(u ε B(u dxdt =. (2.9 Therefore, combining (2.8 with (2.9 gives ( = lim ε + B ε(u ε B(u B ε(u ε B(u dxdt = lim a ε(x,t ε + B ε(u ε 2dxdt, B(u (2.2 where a ε (x,t = [ ( a λ B ε(u ε ( λ B(u + ε dλ, (x,t. Since a ε (x,t is positive and uniformly bounded in, (a ε 2 B ε(u ε 2dxdt B(u C Then, from the definition of a ε, (2.2 and (2.2, lim ε + a ε [ 2dxdt B ε(u ε B(u B ε(u ε B(u 2dxdt. (2.2 EJQTDE, 26 No. 5, p. 7

= lim (a ε + ε 2 B ε(u ε 2dxdt B(u =. This, together with (2.5 yields (2.7, namely B ε(u ε B(u, in L 2 ( as ε +, which deduces (2.6. The proof is complete. In order to reach Theorem.2, we need the following two technical lemmas, which may be found in [9, [4. Lemma 2.4 Let Ω R n be a bounded domain, {u m } be a uniformly bounded sequence in L (Ω and u L (Ω with u m u, wealy in L (Ω as m. ssume that (s,b(s are continuous functions, and (s is nondecreasing. If for any α (R, B( (α contains only a single point, and (u m (x ω(x, a.e. x Ω as m, then (u(x = ω(x, lim B(u (x = B(u(x, a.e. x Ω. m Lemma 2.5 Let Ω R n be a bounded domain and < q <. ssume {f m } is a sequence in L q (Ω and f L q (Ω with f m f, wealy in L q (Ω as m. Then lim f m L q (Ω f L q (Ω. m 3 Proof of the Main Result In this section, we will complete the proof of Theorem.2 based on the estimates and convergence established in 2. Proof of Theorem.2. For any R, any η > and any nonnegative functions ϕ i C ( (i =,2 with suppϕ i [, (,T and according to Proposition 2., H η (u ε, ϕ t dxdt + B ε(u ε ϕ (,t = ϕ 2 (,t, ϕ (,t = ϕ 2 (,t, < t < T, F η (u ε, ϕ dxdt H η (u ε ϕ dxdt EJQTDE, 26 No. 5, p. 8

B ε(u ε H η (u ε u ε + H η(τ (f(τ f(dτ + H η ( ϕ dxdt (ϕ (,t ϕ (,tdt [ (u ε ϕ 2 t + (f(u ε f( ϕ 2 B ε(u ε ϕ2 dxdt =. (3. By the definitions of H η (s, and F η (s,, and by using Lemma 2.4 with (2.2, we get lim H η(u ε (x,t, = H η (u(x,t,, ε + lim F η(u ε (x,t, = F η (u(x,t,, a.e. in. ε + Combining this with (2.6 and (2.2, we have ( lim H η (u ε, ϕ ε + t dxdt + F η (u ε, ϕ dxdt B ε(u ε H η (u ε ϕ dxdt = H η (u, ϕ t dxdt + F η (u, ϕ dxdt B(u H η (u ϕ dxdt (3.2 and lim ε + = [ (u ε ϕ 2 t + (f(u ε f( ϕ ( 2 [ (u ϕ ( 2 t + (f(u f( ϕ 2 B(u ϕ2 B ϕ2 ε(u ε dxdt dxdt. (3.3 Note that u satisfies (.2 and (.3 in the sense of (.4 and (.5. Therefore, owing to (3.2, (3.3 and (3., it is shown that u is just a renormalized solution of the problem (. (.3 provided lim ε + B ε(u ε H η(u ε u ε ϕ dxdt B(u H η(u u ϕ dxdt, (3.4 which will be proved below. Multiplying (2.8 by u ε and then integrating over, we derive that by the formula of integration by parts 2 u 2 ε t dxdt f(u ε u ε dxdt = From (2. and Lemma 2., u 2 ε t dxdt = ( u 2 ε(x,t u 2,ε(x B uε ε(u ε dxdt. (3.5 dx C, EJQTDE, 26 No. 5, p. 9

These two estimates and (3.5 yield f(u ε u ε dxdt sup f(u ε B uε ε(u ε dxdt C. From the definitions of and B ε, the above inequality leads to K ε(u ε p dxdt C, u ε dxdt C. where Thus namely K(u L p ( with K ε (s = s K(s = (b(σ + ε (p /p dσ, s R. K(u p dxdt C, s b (p /p (σdσ, s R. Moreover, (2.2 and (2.4 imply K ε (u ε K(u, wealy in Lp ( as ε +. (3.6 Therefore, for fixed η >, K(u ( H η (u ϕ /p L p (, and (3.6 implies K ε (u ε ( H By Lemma 2.5, K(u η(u ε ϕ /p K(u ( H η(u ϕ /p L p (Q lim ( H η(u ϕ /p, wealy in L p ( as ε +. ε + K ε(u ε ( H η(u ε ϕ /p L p (Q. This is just (3.4. The proof of Theorem.2 is complete. cnowledgment We would lie to express our thans to the referee and the editor. Their comments are very important for the improvement of this paper. EJQTDE, 26 No. 5, p.

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(Received December 3, 25 E-mail addresses: matwzj@jlu.edu.cn yinghua@jlu.edu.cn EJQTDE, 26 No. 5, p. 2