Some asymptotic properties of solutions for Burgers equation in L p (R)

Transcription

1 ARMA manuscript No. (will be inserted by the editor) Some asymptotic properties of solutions for Burgers equation in L p (R) PAULO R. ZINGANO Abstract We discuss time asymptotic properties of solutions u(,t) C ([, [,L p (R)) of Burgers equation corresponding to initial states u L p (R), where p <. As u(,t) L r (R) = O(t ( p r ) ), p r, the products t ( p r ) u(,t) L r (R) are examined and shown to have well defined limit values as t, which are explicitly computed. Some fundamental properties are obtained along the way.. Introduction In this work we discuss a number of properties related to the large time behavior of solutions u = u(x,t) of the parabolic equation u t + au x + buu x = µu xx, x R, t > (a) with initial states in L p (R) for some p <, i.e., u(,) = u L p (R). (b)

2 PAULO R. ZINGANO Here, a, b, µ are given constants, with µ >. Our main concern is the case b (Burgers equation), but, for completeness, b = (heat equation) is also included. Condition (b) is meant in L p sense, i.e., u(,t) u as t ; thus, we are L p (R) interested in solutions u(,t) that stay in L p (R) for t > and are continuous in t in the strong topology, that is, u(,t) C ([,T [,L p (R)) for T >. Such solutions to (a), (b) are actually globally defined (T = ), smooth (C for t > ) and uniquely determined by the initial data, see Section below. Moreover, using the so-called Hopf-Cole transformation [], [], they are explicitly representated as u(x,t) = (x y at ) + 4 µ t e ϕ (y)u 4π µ t ϕ(x, t) (y) dy, (a) where ϕ(x,t) = (x y at ) + 4 µ t e ϕ (y) dy 4π µ t (b) with ϕ L (R) given by the Hopf-Cole transform ϕ (x) = e b x µ u (ξ )dξ. (c) In [], this approach is used to list several solutions u(, t) of particular interest, along with some of their basic features. Here, we take u L p (R) arbitrarily and derive a number of important, general features of the associated solutions u(, t); in most cases, these properties cannot be easily obtained from (a) (c) alone. One example is the fundamental sup-norm estimate u(,t) L (R) C p u L p (R) ( µ t ) p t >, (3) ( p where C p = ) p, so that u(,t) L p (R) L (R) for each t >. p

3 Asymptotic properties of Burgers equation 3 It follows that u(,t) L r (R) C p r p u L p (R) ( µ t ) ( p r ) t > (4) for each p r, because u(,t) u L p (R) for all t, see Section. L p (R) More is true: for any p r, the limit quantities γ p,r = lim ( ) p r t t u(,t) L r (R) (5) are all defined, with γ p,r vanishing for each r whenever p >. The case p = is harder but more interesting: denoting by m the (time invariant) solution mass, i.e., m = + u(x,t) dx = + u (x) dx, (6) it turns out that γ,r = for all r if m =, and γ,r = m r µ (4 µ ) 4π µ ( e µ ) F > (7) L r (R) if m, with F L (R) L (R) defined by F (x) = where erf(x) is the error function e x λ h erf(x), (8a) and λ, h are given by λ = + e erf(x) = π µ x e ξ dξ, h = e µ (8b). (8c) In the important case r = p =, (7) reads lim t u(,t) L (R) = m, (9)

4 4 PAULO R. ZINGANO a result that is well known for the heat equation (see e.g. [7], p.369, or [], p.); the other results for b = are obtained from (7) in the limit as b, giving lim t ( r ) u(,t) L = m 4π µ ( 4π µ r ) r () for heat equation, for every r. Moreover, in the case p =, solutions satisfy an important ergodic property: for any pair u(,t), ũ(,t) C ([, [, L (R)) of solutions of equation (a) carrying the same mass, one has lim t ( r ) u(,t) ũ(,t) L = () for each r, uniformly in r, while, in case of different mass values, one has lim inf t t ( r ) u(,t) ũ(,t) L >. A similar, related result may be used r (R) to identify the parameters a, b, µ in the equation: if u(,t), û(,t) C ([, [, L (R)) are solutions of u t + au x + buu x = µu xx, û t + âû x + ˆbûû x = ˆµû xx, (a) resp., corresponding to initial states u, û L (R) having the same mass m, then the following properties are equivalent to each other: (a, b, µ ) = (â, ˆb, ˆµ ), (b) lim inf t t ( r ) u(,t) û(,t) L r (R) lim t t ( r ) u(,t) û(,t) L r (R) = for some r, (c) = for all r, (d) uniformly in r. These and related results are discussed in more detail in the following sections.

5 Asymptotic properties of Burgers equation 5. Basic properties Before we start the analysis, it will be convenient to review some basic results needed to derive (3) (). Following [8], we introduce a one-parameter family of functions L δ ( ) as follows: taking δ > and S C (R) monotonically increasing with S() =, S(v) = for all v and S(v) = for all v, we set v/δ L δ (v) := δ S(w) dw, v R, (3) so that L δ ( ) approximates the sign function, i.e., L δ (v) sgnv as δ. Theorem. If u(,t) C ([,T [, L p (R))solves problem (a), (b) in [, T [, then u(,t) L p (R) u L p (R) t [,T [. (4) Proof. Taking L δ ( ) as in (3), we multiply (a) by pl δ (u) p L δ (u) and integrate the result on R x[t,t], given < t < t, getting + t L δ (u(x,t)) p dx + µ t + Φ δ (u(x,τ))u x(x,τ) dx dτ = + L δ (u(x,t )) p dx where Φ δ (v) := L δ (v) p. Since Φ δ, this gives u(,t) L p (R) u(,t ) L p (R) as δ, so that, letting t, we obtain (4). Moreover, the L -norm is special in that solutions are L -contractive: Theorem. Let u(,t), ũ(,t) C ([,T [, L p (R))be solutions of equation (a) in the interval [,T [, with initial states u, ũ L p (R). If u ũ L (R), then u(,t) ũ(,t) L (R) u ũ L (R) t [,T [. (5) Proof. Similar to Theorem, see e.g. [8], p. 533.

6 6 PAULO R. ZINGANO We are now in position to obtain (3), extending the L method described in [3]: Theorem 3. If u(,t) C ([,T [, L p (R)) solve problem (a), (b) in [,T [, then u(,t) L (R) C p u L p (R) ( µ t ) p ( p where C p = ) p. p t [,T [ (6) Proof. Multiplying (a) by p(t t )L δ (u) p L δ (u) if p >, or (t t )u(,t) if p =, and integrating over R x [t,t], we obtain, letting δ and t, ( t w(,t) + 4 ) t µ τ w x (,τ) = L (R) p L (R)dτ t w(,τ) L (R)dτ for w(,t) = u(,t) p if p >, w(,t) = u(,t) if p =. Because w(,t) L (R) p 3 3 u(,t) t L p (R) w x(,t) 3 L (R) w(,τ) 4p u L (R)dτ t 3 L p (R) This gives w(,t) L (R) ( p, we get, by (4) and Hölder s inequality, ) u p inequality w(,t) w(,t) L (R) L (R) ( t 3. τ w x (,τ) L (R)dτ) L p (R) ( µ t ) in view of Sobolev w x (,t) L (R), showing (6). It follows from these results that solutions u(,t) C ([,T [, L p (R)) can be continued to [T, [, i.e., they are globally defined. Also, (4) and (6) give (4) by interpolation, so that solutions decay in L r for any r > p. Using standard estimates for fundamental solutions of linear parabolic problems (see [], [9]), one immediately obtains decay estimates for derivatives of u(,t) as well, for example u x (,t) L r (R) C(b, p,r, µ,t ) u L p (R) t ( p r ) t t (7) for each t >, where C(b, p,r, µ,t ) is some positive constant whose value depends on t and the parameters b, p, r, µ, as well as the magnitude of u L p (R). Moreover, Theorem 3 assures uniqueness for strongly continuous solutions:

7 Asymptotic properties of Burgers equation 7 Theorem 4. Let u(,t), ũ(,t) C ([,T [, L p (R)) be solutions of equation (a) corresponding to initial states u, ũ L p (R), respectively. Then u(,t) ũ(,t) L p (R) u ũ L p (R) ek(p,µ)t /p t [,T [ (8) with K(p, µ) = p b C ( p 4 µ +/p max{ u, ũ L p (R) }, C p L p p = ) p. (R) p Proof. The difference θ(,t) = u(,t) ũ(,t) C ([, T [, L p (R)) satisfies θ t + aθ x + b ((u + ũ)θ ) x = µ θ xx, < t < T with θ(,) = u ũ L p (R); multiplying this equation by pl δ (θ) p L δ (θ) and integrating over R x [t, t ], we obtain, letting δ and t, θ(,t) p L p (R) + µ p(p ) t + b p(p ) t + + θ p θ x dx dτ θ(,) p L p (R) + θ p ( u + ũ ) θ x dx dτ, so that θ(,t) p L p (R) θ(,) p + b t L p (R) 8 µ p(p ) σ(τ) θ(,τ) p dτ L p (R) where σ(t) = u(,t) + L (R) ũ(,t). Recalling (4) and (6) above, this L (R) gives (8) by a standard application of Gronwall s lemma. When p =, it is clear that solution u(,t) given in (a) (c) belongs to the class C ([, [, L (R)); for p >, a standard procedure of replacing u L p (R) by cut-off approximations u (l) = u χ [ l, l ] L (R) L p (R), together with the results in this section, show that (a) (c) define u(,t) C ([, [, L p (R)) with initial value u(,) = u, so that it is the solution we seek. These solutions will be further investigated in the next two sections.

8 8 PAULO R. ZINGANO 3. Asymptotic limits: p = In this section, we establish properties (7) () in case of (arbitrary) initial states u L (R). Clearly, by the change of variable ξ = x at we may assume without loss of generality a =, so that u(,t) C ([, [, L (R)) satisfies u t + buu x = µu xx, u(,) = u L (R). (9) A fundamental quantity in this case is given by the solution mass, see (6) above. Before we proceed, it will prove convenient to review the case of heat equation. 3.. Asymptotic limits for heat equation In case b =, u(,t) C ([, [, L (R)) is given by the well known formula u(x,t) = + 4π µ t (x y) 4 µ t e u (y) dy, x R, t >. () We start with the following lemma. Lemma. Let u L (R) have zero mass. Then, for every r, one has uniformly in r. lim t ( r ) u(,t) L =, () Proof. We first show that lim = : given ε >, pick A > so as t u(,t) L (R) u (y) dy ε, so that, from (), we get u(,t) L (R) y A ε + + π e y A ( ξ y 4 µ t ) u (y) dy dξ.

9 Asymptotic properties of Burgers equation 9 Letting t, this gives lim sup t u(,t) L (R) ε + + e ξ π u (y) dy dξ y A ε + u (y) dy ε, y A + where we have used u (y)dy =. Hence, lim t u(,t) L (R) =, as claimed. Next, given < r <, we have, recalling that u(,t) L (R) C(µ) u L (R) t, t ( r ) u(,t) L r (R) u(,t) r L (R) (t r u(,t) L (R) ) C u(,t) r L (R) for some constant C >, so that lim t ( r ) u(,t) L = from the previ- ous case. Finally, for r =, we get, because u x (,t) C(µ) u t 34, L (R) L (R) t u(,t) t u(,t) L (R) L (R) u x (,t) L (R) C(t 4 u(,t) L (R) ) and the result follows from the case r = already considered. Theorem 5. Let u L (R) have mass m R. Then u(,t) given in () satisfies lim t u(,t) L (R) = m, (a) lim t t ( r ) u(,t) L r (R) = m 4π µ ( 4π µ r lim t t u(,t) L (R) = Proof. This is obvious for ũ(x,t) = m 4π µ. r ), < r <, (b) (c) m (x y) 4 µ t e dy, i.e., the solution 4π µ t with the elementary initial data ũ = m χ [,]. The result then follows for arbitrary u L (R) with mass m because lim t ( r ) u(,t) ũ(,t) L = for all r, recalling Lemma above.

10 PAULO R. ZINGANO 3.. Asymptotic limits for Burgers equation We now turn to (9) assuming b. Using the Hopf-Cole transformation [], the solution u(,t) C ([, [, L (R)) of problem (9) can be computed by u(x,t) = µ b ϕ x (x,t) ϕ(x, t) (3) with ϕ(,t) given by ϕ t = µ ϕ xx, t >, (4a) b x µ ϕ(x,) = ϕ (x) e u (ξ )dξ, (4b) so that we have u(x,t) = + (x y) 4 µ t e ϕ (y)u 4π µ t ϕ(x, t) (y) dy. (5) As before, we start with the following lemma. Lemma. Let u L (R) have zero mass. Then, for every r, one has uniformly in r. lim t ( r ) u(,t) L =, (6) Proof. From (4), we see that ϕ x (,t) satisfies the conditions of Lemma above, so that, for every r, we have lim t ( r ) ϕx (,t) =. Since, t Lr(R) by Theorem, /ϕ(,t) is uniformly bounded, we obtain (6) from (3). Theorem 6. Let u(,t), ũ(,t) C ([, [, L (R)) be solutions of Burgers equation having the same mass. Then, for every r, one has uniformly in r. lim t t ( r ) u(,t) ũ(,t) L r (R) =, (7)

11 Asymptotic properties of Burgers equation Proof. Letting ϕ(,t), ϕ(,t) be the Hopf-Cole transforms of u(,t), ũ(,t), respectively, and setting ω(,t) = ϕ x (,t) ϕ x (,t), we have that ω(,t) has zero mass and satisfies ω t = µ ω xx, so that lim t ( r ) ϕx (,t) ϕ t x (,t) = Lr(R) by Lemma, for every r, uniformly in r, that is, lim t ( r ) ϕ(,t)u(,t) ϕ(,t)ũ(,t) L =, uniformly in r. Since /ϕ(,t), / ϕ(,t) are bounded uniformly in t and ϕ(,t) ϕ(,t) tends to zero as t, we get the result. L (R) We are now in position to compute the limits γ,r for arbitrary u in L (R). Theorem 7. Given u L (R), the solution u(,t) C ([, [, L (R)) of (9) satisfies lim t u(,t) L (R) = m, (8a) lim t t ( r ) u(,t) L r (R) = < r <, lim t u(,t) = t L (R) m (4µ ) 4π µ r µ ( e µ ) F L r (R), (8b) m µ 4π µ ( e µ ) F, (8c) L (R) where F L (R) L (R) is given in (8), and m is the solution mass given in (6). Proof. Recalling Lemma, we may assume m. By Theorem 6 it is sufficient to show the result for the particular initial state u = m χ [,], in which case u(,t) is given by u(x,t) = m 4π µ t ϕ(x, t) (x y) 4 µ t e ϕ (y) dy, (9)

12 PAULO R. ZINGANO where ϕ(x,t) = + 4π µ t (x y) 4 µ t e ϕ (y) dy, b µ ϕ (x) = e x u (ξ )dξ. In particular, for any t >, u(,t) does not change sign, so that u(,t) = L (R) + + u(x,t) dx = u (x) dx = m for all t >, which shows (8a). To obtain (8b) and (8c), we introduce if x < α H (x) = µ e if x > α (3) with < α < chosen so that + (H (x) ϕ (x)) dx =, (3a) i.e., µ α + ( α ) e = µ ( e µ ), (3b) as illustrated in the picture below. H ϕ µ e H - - α x Figure : H and ϕ Setting H (x,t) = + 4π µ t (x y) 4 µ t e H (y) dy, (3)

13 we obtain Asymptotic properties of Burgers equation 3 lim t ( r ) H (,t) ϕ(,t) L = for every r, by (3a) and Lemma, so that we have for ω(, t) defined by lim t ( r ) u(,t) ω(,t) L = ω(x,t) = m 4π µ t H (x,t) with H (x,t) given in (3) above, that is, (x y) 4 µ t e H (y) dy H (x,t) = λ h erf( x α 4 µ t ), (3) where λ, h, erf(x) are given in (8). We can finally derive (8b), for < r < : given ξ R, we have ω (α + ξ 4 µ t, t ) = and, observing that e ( ξ + m e 4π µ t λ herf(ξ) ( ξ + α y 4 µ t ) H (y) dy, α y 4 µ t ) H (y) dy r e r ξ + r H r L (,) for all ξ R and t /4µ, we get, by Lebesgue s theorem, lim t ( r ) ω(,t) L = = m ( r ) + e (4 µ ) H 4π µ (y) dy ξ λ herf(ξ) r dξ = m (4 µ ) 4π µ r µ ( e µ ) F L r (R) in view of (3), (3b). This shows (8b) for arbitrary < r <. /r

14 4 PAULO R. ZINGANO Finally, for r =, we get, from the expression for ω(α + ξ 4 µ t, t) above, lim inf t ω(,t) m µ t L (R) 4π µ ( e µ ) e ξ λ herf(ξ) for every ξ R, so that we have lim inf t ω(, t) m µ t L (R) 4π µ ( e µ ) F. (33a) L (R) On the other hand, for t > let ξ t R be such that ω(,t) L (R) = ω(α + ξ t 4 µ t, t ) ; since lim inf t ω(, t) > from (33a), we must have ξ bounded for all t. t L (R) t Now, given any sequence t n such that ξ n ξ tn converges, say ξ n ξ, we then have tn ω(,t n ) = L (R) so that lim n tn ω(,t n ) = L (R) m e 4π µ λ herf(ξ n ) ( ξ n + m µ 4π µ ( e µ ) α y 4 µ tn ) H (y) dy, e ξ λ h erf(ξ ). This gives lim sup t t ω(,t) m µ L (R) 4π µ ( e µ ) F L (R) (33b) which, together with (33a) above, shows (8c), as claimed. In particular, we may use solutions (with nonzero mass) to determine the coefficients b, µ in equation (a): once two or more quantities γ have been assessed,,r the relations (8b), (8c) can be numerically inverted to obtain b, µ. The large time behavior of solutions can also be used to give out the value of the linear coefficient a in (a), as the next result shows.

15 Asymptotic properties of Burgers equation Asymptotically equivalent systems We now examine when it is that solutions from possibly distinct equations (a) (with nonzero mass) become so approximately close as t that (7) will hold. Thus, we consider a pair of equations u t + au x + buu x = µu xx, û t + âû x + ˆbûû x = ˆµû xx (34) where a, b, µ, â, ˆb, ˆµ are real constants, with µ, ˆµ >, and proceed to show (): Theorem 8. Let u(,t), û(,t) C ([, [, L (R)) be solutions of equations (34), resp., corresponding to initial states u, û L (R) with the same mass m. Then the following statements are equivalent to one another: (a, b, µ ) = (â, ˆb, ˆµ ), (35a) lim inf t ( r ) u(,t) û(,t) L = for some r, (35b) lim ( r ) u(,t) û(,t) L = for all r, (35c) uniformly in r. Proof. Recalling Theorem 6, it is sufficient to consider the case u = û = m χ [,]. If a â, then from (a) (c) there exist constants K, κ > such that u(ξ t + ât, t ) K e 8µ (a â) t κ, û(ξ t + ât, t ) t t for all ξ and t. This clearly gives u(,t) û(,t) L r (R) κ t ( r ) t t for all r, for some t > sufficiently large that depends on m, µ, a â and the magnitude of u, û L. Assuming now that a = â, suppose (R) L (R)

16 6 PAULO R. ZINGANO we have (b, µ ) ( ˆb, ˆµ ): from (7), () we can find < r < such that the limits (5) corresponding to u(,t), û(,t) are different, i.e., γ r ˆγ r, where γ r = lim t ( r ) u(,t) L, ˆγ r = lim t ( r ) û(,t) L for every r. In particular, we get lim inf t t ( r ) u(,t) û(,t) L r (R) γ r ˆγ r >. (36a) Given r > r, we have, by interpolation, u(,t) û(,t) L r (R) u(,t) û(,t) r r r L (R) r u(,t) û(,t) r r L r (R) r r for every t >. This gives, from (36a), lim inf t t ( r ) u(,t) û(,t) L r (R) C γ r ˆγ r ( ) r r r (36b) ( r ) r r where C = (γ + ˆγ ). Similarly, for r < r, we get r r r u(,t) û(,t) L r u(,t) û(,t) u(,t) û(,t) r (R) L r (R) L (R), which gives, by (36a), lim inf t ( r r ) r u(,t) û(,t) L γ ˆγ r r (γ + ˆγ ) r r. (36c) Hence, in all cases above, (a, b, µ ) (â, ˆb, ˆµ ) gives, for every r, lim inf t t ( r ) u(,t) û(,t) L r (R) > ; together with Theorem 6, this completes the argument.

17 Asymptotic properties of Burgers equation 7 In a similar way, we can show that, given initial states u, ũ L (R) with different mass values, the corresponding solutions u(,t), ũ(,t) C ([, [, L (R)) of equation (a) satisfy, for each r, u(,t) ũ(,t) L r (R) c r t ( r ) (37) for all t > large, where c r is some positive constant; hence, the zero limit in (7) can only hold for solutions carrying the same mass. Moreover, it follows from [4], Theorem 3.3, that property (7) also holds for solutions u(,t), ũ(,t) C ([, [, L (R)), with same mass values, of equations u t + f (u) x = (k(u)u x ) x, ũ t + f ()ũ x + f ()ũũ x = k()ũ xx (38) where f ( ), k( ) are smooth (C ), with k(u) bounded below from zero. This result illustrates a fundamental feature of Burgers and heat equation: besides their role in modeling many significant physical phenomena directly, they also provide important approximations to more complex models, see e.g. [], [3], [4], [5], [6], [], [] and references therein. 4. Asymptotic limits: p > We now turn to the case p > and the corresponding time asymptotic behavior of solutions u(,t) C ([, [, L p (R)) of equation (a). Theorem 9. Given p > and u L p (R), the solution u(,t) C ([, [, L p (R)) of problem (a), (b) satisfies lim u(,t) =. (39) t L p (R)

18 8 PAULO R. ZINGANO Proof. Given ε >, let R > be large enough so that x R u (x) p dx ε p. Writing u(,t) = v(,t) + w(,t) where v(,t) C ([, [, L p (R)) is the solution of (a) with v(,) = u ( χ [ R, R ] ), we get v(,t) L p (R) v(, ) L p (R) ε for all t >, by Theorem, while we have w(,t) L p (R) C R w(,t) p L (R) p by Theorem, where C R = u χ. Since w(,t) = [ R, R ] L (R) L (R) O(t p ) by Theorem 3, we get w(,t) ε for all t > sufficiently large if p >. L p (R) Thus, for any u L p (R), u(,t) always decreases monotonically to L p (R) zero when p >. This decay, however, can be arbitrarily slow, so that no rates can be given in general, as in the familiar case of heat equation. Theorem. Given p > and u L p (R), the solution u(,t) C ([, [, L p (R)) of problem (a), (b) satisfies, for every p r, uniformly in r. lim t ( p r ) u(,t) L =, (4) Proof. For r = p, this result was obtained in (39), Theorem 9; for r =, because t p u(,t) L (R) p p u(,t) p L p (R) ( t u x (,t) L p (R) ) p for all t >, the result follows from (7), (39). Finally, given p < r <, we have t ( p r ) p p u(,t) u(,t) r ( t p u(,t) L L r (R) L p (R) (R) ) r and the result follows from the cases r = p, r = already considered. Hence, for p > solutions always decay faster than the rates indicated in (4), but again no better rates can be given for general u L p (R).

19 Asymptotic properties of Burgers equation 9 Acknowledgements. This work was partially supported by CNPQ and FAPERGS, Brazil. References. D. G. ARONSON: Bounds for the fundamental solution of a parabolic equation. Bull. Amer. Math. Soc. 73, (967).. E. R. BENTON AND G. W. PLATZMAN: A table of solutions of the one-dimensional Burgers equation. Quart. Appl. Math. 3, 95 (97). 3. I. L. CHERN: Multiple-mode diffusion waves for viscous nonstrictly hyperbolic conservation laws. Comm. Math. Phys. 38, 5 6 (99). 4. I. L. CHERN AND T. P. LIU: Convergence to diffusion waves of solutions for viscous conservation laws. Comm. Math. Phys., (987). 5. M. ESCOBEDO AND E. ZUAZUA: Large time behavior for convection-diffusion equations in R n. J. Func. Anal., 9 6 (99). 6. C. A. J. FLETCHER: Burgers equation: a model for all reasons. In: J. Noye (Ed.), Numerical Solutions of Partial Differential Equations, North-Holland, New York, 98, S. R. FOGUEL: On iterates of convolutions. Proc. Amer. Math. Soc. 47, (975). 8. E. HARABETIAN: Rarefactions and large time behavior for parabolic equations and monotone schemes. Comm. Math. Phys. 4, (988). 9. A. M. ILIN, A. S. KALASHNIKOV AND O. A. OLEINIK: Second order linear equations of parabolic type. Russ. Math. Surv. 7, 43 (96).. T. P. LIU AND Y. ZENG: Large time behavior of solutions for general quasilinear hyperbolic-parabolic systems of conservation laws. Mem. Amer. Math. Soc., no. 599, Providence, D. S. ORNSTEIN: Random walks, I. Trans. Amer. Math. Soc. 38, 43 (969).. G. B. WHITHAM: Linear and nonlinear waves. Wiley, New York, 974.

20 PAULO R. ZINGANO 3. P. R. ZINGANO: Nonlinear L stability under large disturbances. J. Comp. Appl. Math. 3, 7 9 (999). 4. P. R. ZINGANO: Asymptotic behavior of the L norm of solutions to nonlinear parabolic equations. Comm. Pure Appl. Anal. 3, 5 59 (4). Departamento de Matemática Pura e Aplicada Universidade Federal do Rio Grande do Sul Porto Alegre, RS 95, Brazil pzingano@mat.ufrgs.br