STRONGLY DAMPED SEMILINEAR EQUATIONS

A class of strongly damped semilinear equations is studied by using the theory of analytic semigroups. Conditions (on the nonlinear forcing term) are given under which the existence and uniqueness of local and global classical solutions are ensured.


Introduction
Let fl be a bounded domain in n with sufficiently smooth boundary Of] and let Lu- aa(x)Dau be a strongly elliptic differential operator of order 2m in f.We consider the following strongly damped wave equation utt + (aL + b)u + (cL + d)u f(t, u, ut) in f (to, T), to) x 0 and ut(x to) x 1 for x E , Dau O for (x,t) E OfX[to, T), cl _<m-l,   (1.1) where a > 0. Duvaut and Lions [4], Glowinski, Lions and Tremolieres [5] have studied particular cases of (1.1) in which the n-dimensional Laplacian, L -A, in the context of the theorem of viscoelastic materials.For more information on these particular cases, we refer the reader to Webb [8] and Ang and Dinh [1].
conditions. in a Banach space X under the assumptions that the linear operators A and B can be decomposed as -A-A 1 +A 2 and B-A2A1, where A k generates a c0-semigroup Tk(t), k--1,2,; and f satisfies a locally Lipschit condition.He established the local existence and uniqueness of a mild solution to (1.a) which reads as follows.There exists a continuous function u on [0, c] for some c > 0 such that u satisfies the integral equation u(t) Tl(t) + / TI(t-v)T2(v)( Ale)dr Aviles and Sandefur [2] studied the wellposedness of (1.3) under similar We notice that, for the existence and uniqueness of solutions to (1.2), it suffices to study the problem u"(t) + Au'(t) f(t, u(t), u'(t)), u(tO) X O, (to) Xl, where the terms bu'(t) and (cA + dI)u(t)are merged with f (F).so that f still satisfies assumption Engler, Neubrander and Sandefur [3] proved the local existence and uniqueness of a mild solution to (1.4) under the assumptions that -A generates an analytic semigroup T(t) in X and f satisfies a condition similar to assumption (F), where a mild solution on [t0, tl) for some t 1 > to, to (1.4) is the first component of a solution (u(t), v(t)) of the integral equations u(t) x o + (T(t to)-I)( A)-ix 1 + / (T(t s) I)(-A)-If(s, u(s), v(s))ds, t o < t < t 1 o v(t)-T(t-to)x 1 + / T(t-s)f(s, u(s), v(s))ds, t o <_ t < t 1. o (1.5) We improve the result of [3] by showing that (1.4) has a unique classical solution locally, i.e., there exists a unique u_Cl([to, tl):X)VIC2((to, tl):X) and satisfies (1.4) on [t0, tl) for some t I > t 0. Further, we discuss the continuation of this solution, maximal interval of existence and the global existence.We achieve these objectives by extending the ideas and techniques used in the proofs of Theorems 6.3.1 and 6.3.3 in Pazy [6] concerning the semilinear equations of the first order to problem (1.4).For the global existence, we require a modified version of Lemma 4.1, stated and proved at the end of Section 4 originally stated in Pazy [6] as Lemma 5.6.7.

Preliminaries and Assumptions
Let X be a Banach space and let -A.generate the analytic semigroup T(t) in X.We note that if -A is the infinitesimal generator of an analytic semigroup then -(A + hi) is invertible and generates a bounded analytic semigroup for a > 0 large enough.This allows us to reduce the general case, in which -A is the infinitesimal generator of an analytic semigroup, to the case where the semigroup is bounded and the generator is invertible.Hence, for convenience, without loss of generality, we assume that T(t)is bounded, that is liT(t)II < M for t> 0 and 0 E p(-A), i.e., -A is invertible.Here p(-A) is the resolvent set of -A.It follows that, for 0 _< a < 1, A s can be defined as a closed linear invertible operator with its domain D(A) being dense in X.We denote by X a the Banach space D(A) equipped with the norm I I x I I a I I Aax I I which is equivalent to the graph norm of As.For 0 < a </, we have X C X a and the embedding is continuous.
Regarding the function f we make the following Assumption (F): Let U be an open set of + X 1 X a.A function f is said to satisfy the assumption (F) if for every (t,x,) U there exist a neighborhood Y C U and constants L > 0, By a local classical solution to (2.1) we mean a function u cl([to, tl): X) N C2((to, tl)" X) satisfying (2.1) on [to, t1) for some t 1 > t o.By a local mild solution to (2.1) we mean the first component of a solution (u, v) to the integral equations (1.5) on [to, tl) for some t I > t o.

Local Existence
As we have already pointed out, without loss of generality, the semigroup, generated by -A, can be assumed to be bounded and A invertible.Under these conditions imposed on A we prove the following local existence and uniqueness theorem.Theorem 3.1" Suppose that -A generates the analytic semigroup T(t) such that [i T(t) I] <-M and 0 p(-A).If the map f satisfies assumption (F) then (2.1) has a unique local classical solution.
Proof: Fix (to, Xo, Xl) in U and choose t] > o and 5 > 0 such that (2.2), with some fixed con- stants L >_ 0 and 0 < t9 _< 1, holds on the set (3.1) Let Bmax I I f(t, xo, Xl)I1" Choose t I > t o such that o _< _< t 1 t I t o < min{t to,(1 where C a is a positive constant depending on c and satisfying I I AaT(t) I I <-Cat-o, for t > 0.
For every y Y, Fy(to)-(Azo, AaXl), and the assumption (F) on f implies that F:Y-oY.
Hence F:S-S.Now let (Yl, Y2) and (Zl, Z2) be any two elements of S. We use assumption (F)   to get I I if1(t) -'1(t) I I + I I 2(t) -2(t) I I Thus F is a contraction on S. Therefore, it has a unique fixed point in S. Let (ffl, 2) S be that fixed point of F.Then, l(t) Ax o (T(t to) I)x 1 / (T(t s) I)f y (s)ds, (3.5) o y2(t) T(t to)Aaxl + / T(t s)Aaf y (s)ds, o where f-(t)-f(t,A-lyl(t), A-a2(t)).We note that (u,v)-(A-II,A-a2) is the unique solution of integral equations (1.5) on [t0, tl].Assumption (F) and the continuity of Yl and Y2 on [to, tl] imply that the map t-f-(t) is continuous and hence bounded on [to, tl].Let I I fy (t)II -N for t o _ t _< t.We will now show that tf-(t) is locally HSlder continuous on (to, tl].For this we first show that 1 and 2 are locally HSlder continuous on (t0, tl].From Theorem 2.6.13 in Pazy [6], for every 0 < fl < 1-c and every 0 < h < 1, we have I I (T(h) I)Aa(t s)In I I Aa + fT(t s)In C/H( t s)-( + f).
Here M 1 depends on t and increases to infinity as tto, while M 2 and M 3 can be chosen indepen- dent of t.From the above estimates, it follows that there exists a positive constant C such that for every t o > to, I I 2(t) 2()II C It s for t o < t _< t, s <_ t 1.
Similar result holds for Yl (if we take a-0 in the above considerations).The local HSlder contin- uity of f_-.(t) on (to, t] follows from assumption (F) and the local HSlder continuity of 1 and Y2 on (to, tl. Consider the initial value problem -t(t) + Av(t) f (t) (3.7) V(to) X 1.
For t > to, each term on the right-hand side belongs to D(A) and hence belongs to D(Aa).
Remark 3.1: Theorem 3.1 can be applied to assert that a unique local mold solution of the general linear second order equation established by Engler, Neubrander and Sandefur [3] in Example 3.1, is in fact a local classical solution.Using arguments of [3], this solution can be proved to be global.
Itemark 3.2: Theorem 3.1 can also be used to prove the regularity of local mild solutions to the strongly damped quasilinear wave equations and the strongly damped Klein-Gordon equation of Examples 3.2 and 3.3 in [3], respectively.

Global Existence
In [3] it was proven that if f: [0, xz) X 1 X o.--X satisfies assumption (F) globally, then the solution (u,v) to integral equations (1.5) can be continued to the maximal interval of existence [0, T) and, if T < , then lim [11 u()I1 I I v()II 3-/ , tT We prove the following global existence result.
Theorem 4.1: Let 0 D(-A) and let -A be the infinitesimal generator of an analytic semi- group T(t) such that I I T(t)II <_ M fo t >_ O, Lt f: [0, ) Xa X--,X satisfy assumption (F).
If there exists a nondecreasing function k:[t0, c)R+, such that I I f(t,, )11 < k(t)[1 + I I x II1 -4-I I I I ] for t > o, (,) X X, then for each (x0, Xl) X 1 x Xa, (2.1) has a unique classical solution u which exists for all t > t o.
Proof: Let [t0, T be the maximal interval of existence for the solution u to (2.1) guaranteed by Theorem 3.1.It suffices to prove that [11 (t)I11/ I I v(t)II ] _< c on [t0, T for some xed constant C > 0 independent of t.Now, since u(t)is a solution of (2.1) on [t0, T), it is also a mild solution to (2.1).Therefore, from (3.5), we have Au(t) Az o -(T(t to) I)x I (T(t s) I)f (s)ds, o T(t-to)Aaxi + / T(t-s)aa? (s)ds, Aau'(t) o where f (t)-f(t, u(t), u'(t)) for t [to, T).Our assumptions on f imply that [1 + I I u(t)II1-4-I I '(t)II 1 Strongly Damped Semilinear Equations 403 --( C1 4-C2 i [1 + I I ()Ill + [I u'()II ]d o + C3 i ( s)-[1 + I I u(s)111 + I I u'()II ]d.o Now, we only need to apply the following lemma to get the required estimate.Lemma 4.1: Lel (t,s) >_ 0 be continuous on 0 <_ s < t <_ T. If there are positive constants A, B1, B 2 and fl, such that 8 for 0 < s < t < T, then there exists a positive constant C such that (t,s) < C for 0 < s < t < T.
Proof of Lemma 4.1: We have T and the well-known identity t r) l(r 7")"'-1do" (t r) 1 + lr(5) r(7) (3.10) (3.11) get Iterating (3.9) n-1 times using (3.10) and (3.11) and majorating (t-s) and (t-r) by Choosing n sufficiently large so that n > 1 and replacing (t-r) n -1 by T n -1 we get (t, S) _< C 1 4-c 2 / (, s)do', 8 where c 1 and c 2 are positive constants independent of s.The required result then follows from the Gronwall's inequality.This ends the proof of the lemma and of Theorem 4.1.

Remark 4. 1 :
Theorem 4.1 gives the global existence and uniqueness of a classical solution to the Klein-Gordon equation considered in Example 3.3 in [3] (cf. also Example 4.1 in [3]).