zbMATH — the first resource for mathematics

Gevrey well-posedness of an abstract Cauchy problem of weakly hyperbolic type. (English) Zbl 0706.35077
The main result presented in this paper is a global existence theorem for solutions to a general second order Cauchy problem of weakly hyperbolic type which is studied in an abstract Hilbert space setting. More precisely, let (H,$$| \cdot |)$$ be a Hilbert space, $$V\subset H$$ a dense subset, $$B:=(b_ 1,...,b_ n)$$, $$n\in {\mathbb{N}}$$, an n-tuple of commuting densely defined closed linear operators $$b_ i\in {\mathcal L}(V,H).$$
For fixed $$s\in R$$, $$s>1$$, $$B=(b_ 1,...,b_ n)$$ generates the Banach scale of generalized Gevrey spaces $$X^ s_ r(B).$$
Let $$T\in R^+$$, $$I:=[0,T]$$, $$k\in]0,2[$$, $$A\in C^ k(I;{\mathcal L}(V,V'))$$ with $$<A(t)v,w>=<A(t)w,v>$$, v,w$$\in V$$, $$M\in L^ 1(I;{\mathcal L}(V,H))$$, $$f\in L^ 1(I;H)$$ and $$s\in [1,1+(k/2)[.$$
In case $$s>1$$ the additional assumption is made that H possesses a countable basis consisting of common eigenvectors of the operators $$b_ i$$, $$i=1,...,n$$. Under these hypotheses and the weak hyperbolicity condition $$<A(t)v,v>\geq 0$$, $$v\in V$$, the author proves that for sufficiently small $$r_ 0$$ and any $$u_ 0,u_ 1\in X^ s_{r_ 0}(B)$$ there exists a unique smooth solution u: $$I\to X^ s_{\bar r}$$ for some $$\bar r\in]0,r_ 0]$$ depending on T to the Cauchy problem $(C)\quad u''+[A(t)+M(t)]u=f(t),\quad t\in I,\quad u(0)=u_ 0,\quad u'(0)=u_ 1$ an analogous existence and uniqueness result for strongly hyperbolic equations is also derived.
The proof depends in an essential way on certain a priori estimates for the “Gevrey-type infinite order energy function” of the solution to (C); the author first proves existence of solutions $$u_ N$$ to a sequence of approximating Cauchy problems $$(C_ N)$$ and then by applying the a priori estimates to these solutions he obtains a uniform bound for the infinite order energy functions of $$u_ N$$; a compactness argument yields the desired result. As an application of his abstract theorems the author re-proves some global and local existence results found earlier by Jannelli and Nichitani in a different approach to the Cauchy problem for weakly and strongly hyperbolic second order equations [see E. Jannelli, J. Math. Kyoto Univ. 24, 763-778 (1984; Zbl 0582.35070), and T. Nichitani, Bull. Sci. Math., II. Ser. 107, 113-138 (1983; Zbl 0536.35042)]. Apart from this some applications to certain concrete differential operators are also considered.
Reviewer: H.-J.Böttger

MSC:
 35L15 Initial value problems for second-order hyperbolic equations 34G10 Linear differential equations in abstract spaces 35B45 A priori estimates in context of PDEs 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
Full Text:
References:
 [1] Arosio, A. and Spagnolo, S., Global existence for abstract evolution equations of weakly hyperbolic type, J. Math, pures et appl., 65 (1986), 263-305. · Zbl 0616.35049 [2] Cardosi, L,., Evolution equations in scales of abstract Gevrey spaces, Boll. UMI, An. Funz. e Appl., 6, 4-C, n. 1 (1985), 379-406. · Zbl 0612.35120 [3] Colombini, F., de Giorgi, E. and Spagnolo, S., Sur les equations hyperboliques avec des coefficients qui ne dependent que du temps, Ann. Sc. Norm. Sup. Pisa, 6, 3 (1979), 511-559. · Zbl 0417.35049 · numdam:ASNSP_1979_4_6_3_511_0 · eudml:83819 [4] Colombini, F., Jannelli, E. and Spagnolo, S., Well-posedness in the Gevrey classes of the Cauchy problem for a non-strictly hyperbolic equation with coefficients depending on time, Ann. Sc. Norm. Sup. Pisa, 10, 2 (1983), 291-312. · Zbl 0543.35056 · numdam:ASNSP_1983_4_10_2_291_0 · eudml:83908 [5] D’Ancona, P., Global solution of the Cauchy problem for a class of abstract non- linear hyperbolic equations, to appear on Ann. Mat. pura ed appl. · Zbl 0683.35053 · doi:10.1007/BF01762786 [6] Glaeser, G., Racine carree d’une fonction differentiable, Ann. Inst. Fourier, 13 (1963), 203-210. · Zbl 0128.27903 · doi:10.5802/aif.146 · numdam:AIF_1963__13_2_203_0 · eudml:73807 [7] Jannelli, E., Gevrey well-posedness for a class of weakly hyperbolic equations, J. Math. Kyoto Univ., 24 (1984), 763-778. · Zbl 0582.35070 [8] , Weakly hyperbolic equations of second order with coefficients real analytic in space variables, Comm. PDF, 7 (1982), 537-558. · Zbl 0505.35051 · doi:10.1080/03605308208820231 [9] Lions, J. L., and Magenes, E., Non-homogeneous boundary value problems and applications, (Springer, Berlin, 1973). · Zbl 0251.35001 [10] Nishitani, T., Sur les equations hyperboliques a coefficients qui sont holderiens en t et de classe de Gevrey en x, Bull. Sci. Math. 2e serie, 107 (1983), 113-138. · Zbl 0536.35042 [11] Ohya, Y. and Tarama, S., Le probleme de Cauchy a caracteristiques multiples dans la classe de Gevrey (coefficients holderiens en t), Proc. of the Taniguchi International Symposium on hyperbolic equations and related topics, Katata and Kyoto 1984, S. Mizohata ed., (Kinokuniya, Tokyo, 1986), 273-306. · Zbl 0665.35045 [12] Oleinik, O.A., On linear equations of second order with non-negative characteristic form, Mat. Sb. N.S., 69 (111) (1966), 111-140 (transl.: Transl. Amer. Mat. Soc. (2) 65, 167-199). [13] Spagnolo, S., Global solvability in Banach scales of weakly hyperbolic abstract equations, in Ennio de Giorgi Colloquium, P. Kree ed., Research Notes in Math., 125, (Pitman, Boston, 1985), 149-167. · Zbl 0582.35072
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.