ON WEAK SOLUTIONS OF SEMILINEAR HYPERBOLIC-PARABOLIC EQUATIONS

In this paper we prove the existence and uniqueness of weak solutions of the mixed problem for the nonlinear hyperbolic-parabolic equation (K1(x,t)u′)′


INTRODUCTION
In this paper we study the global existence and uniqueness of weak solutions to the mixed problem for the nonlinear hyperbolic-parabolic equations (K(x,t)u')' + K(x,t)u' + A(t)u + F(u) f in Q, (P) u(x,O) uo(x), u'(x,O) u(x), x where Q is a cylindrical domain of R n+l and Kl(z,t), K2(z,t), F are functions which satisfy some appropriate conditions Physical motivations for studying (P) come from several problems of continuum mechanics, such as turbulence, combustion, material aging, transonic flows, etc Let f be a bounded open set in R By Q we represent the cylinder f2 ]0, T[, T an arbitrary positive real number In Q we consider the mixed problem for the hyperbolic-parabolic equation KI where K1 (x) _> 0 and Kg.(x) >_ fl > O, x E f This type of equation was studied by Bensoussan-Lions-Papanicolau in Medeiros [2] studied the existence of weak solution of the mixed problem for (1 1) plus the nonlinear term lulvu, p > 0 Lima [3] analyzed the equation (1 1) in a nonlinear abstract framework In Lar'kin [4] (1 1) was studied with more general nonlinearities, K1 and Kg.depends also t, included also in f, but still with null initial conditions, plus strong restrictions on f Many authors studied the equation (1 1) when coefficients K and K9. also depend on Among them we mention Bryukhanov [5], Bubnov  [6], Vragov [7] and Gadzhier [8] All of them assume zero initial data FI:,I,RI';IRA A significant nonlinear generalization of problem (1 1) is the following (K(z,t)u') + K.,.(z,t)u' Au + F(u) f in Q with initial data (z, 0) 0(:),,'(z, 0) (z), ( (1 3) Strauss [8] studied the existence of weak solution for (1 2) and (1 3) when K 1, K.,, 0 and F is a function that satisfies F continuous and sF(s) _> 0 for all s in R.

4)
Maciel [10] studied existence and uniqueness of weak solutions problem (1 2)-( 13), when F is continuous and sF(s) _> 0 for all s E R, where the uniqueness is proved only for some particular cases of function F, and K1 and K2 satisfies

But with null initial conditions
The problem (1 2) may be included in the following general formulation ( ( Observe that on the set K (x, t) 0 the equation (1 7) degenerate into parabolic equation In this paper we study existence and uniqueness of weak solution of the mixed problem for the equation (1 7) in the case of null initial data, with F satisfying condition (1 4) For the existence we apply the Faedo-Galerkin method (see Lions 11 ]), a priori estimates not usual and a result ofW A Strauss for the nonlinear term (see Strauss [9]).The uniqueness is considered only for some particular cases of F which permit the application of a method due to Visik and Ladyzenskaya 12] The paper is organized as follows.
2 Some terminology and assumptions.
Existence of weak solutions 4 Uniqueness 2. SOME TERMINOLOGY AND ASSUMPTIONS By D(f) we denote the space of infinitely differential functions with compact support contained in f, the inner product and norm in L2(f) and H0(f2) will be represented by (.,.), I.I and ((.,.)), I1.11 respectively By H-(f) we denote the dual space of H (f) Let X be a Banach space, we denote by L'(0, T; X), 1 < p < c, the Banach space of vector-valued functions u-(0, T) X which are measurable and Ilu(t)ll, L'(0, T) with the norm IlUllr.oo(O.T:X) ess sup Ilu(t)llx.0<t<T Let us consider the following family of operators in L(Hlo (fl), H -1 (f)) where az Here t denote the derivative in distributional sense for all i,.We suppose that E a:3(x' t)t3 J([l[2 + + [n [2) t,3= 753 (23)   for all (t, ) [0, T] x R and a.e. in f, with > 0 a constant.
If we denote by a(t, u, v) the family ofbilinear forms in H(f) x H0 (f) associated with A(t), we have t) wch is setc.From (2.3) it follows that a(t, u, u) llull u, for 1 u Hd() d [0, T].
(3.11) PROOF.We know (see [9]) tt there ests a sequence of nctions F R R inch that each F is Lipsct with constt ak, derivable except on a fite number of points, sF(a) 0 d the sequence converges ufoy to F on the bounded sets of R.
PROOF.With sF(s) _ 0 and F is continuous then F(O) 0 and f F()d( _ 0 Then F satisfies conditions ofMello 13 ], and therefore we have the uniqueness.THEOREM 4.3.Suppose that F R -R is a global Lipschitz function such that sF(s) _> 0 for all s 6 R Then exists a unique function (z, t), (z, ) Q that is a solution of the problem (3.5)- (3.8).PROOF.See Maciel 10]
OF SEMILINEAR HYPERBOLIC-PARABOLIC EQUATIONS