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On a global in time existence theorem of smooth solutions to a nonlinear wave equation with viscosity. (English) Zbl 0788.35001
We prove the global in time existence of smooth solutions to a nonlinear wave equation with linear viscosity describing the motion of a fixed membrane with viscosity for any smooth initial data. In our approach, the key is to prove the local in time existence of smooth solutions to the nonlinear wave equation with linear viscosity of the form: \(u_{tt}- A(u)-Bu_ t=0\), in a bounded or unbounded domain with zero Dirichlet condition, where \(A(u)\) is a quasilinear operator of second order and \(B\) is a linear elliptic operator of second order. To solve the problem, we can not use the fractional power of the operator \(B\) with zero Dirichlet condition, because its domain is not clear. We use the following simple linearization procedure. Differentiation in the time variable \(t\) yields the equation \(v_{tt}-A'(u) v-Bv_ t=0\) for \(v=u_ t\). This is combined with the equation \(v_ t-A(u)- Bu_ t=0\). Applying \(L^ p\) theories of linear elliptic equations and linear parabolic equations to the operator \(B\) and to the operator \(\partial_ t-B\), respectively, and using the usual successive approximation, our local in time existence theorem is proved.

MSC:
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35L70 Second-order nonlinear hyperbolic equations
74D99 Materials of strain-rate type and history type, other materials with memory (including elastic materials with viscous damping, various viscoelastic materials)
35L20 Initial-boundary value problems for second-order hyperbolic equations
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[1] Agmon, S.: On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems. Commun. Pure Appl. Math.15, 119-147 (1962) · Zbl 0109.32701 · doi:10.1002/cpa.3160150203
[2] Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I. Commun. Pure Appl. Math.12, 623-727 (1959); ibid, Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II. Commun. Pure Appl. Math.17, 35-92 (1964) · Zbl 0093.10401 · doi:10.1002/cpa.3160120405
[3] Andrews, G.: On the existence of solutions to the equation:u tt =u xxt +?(u x ) x . J. Differ. Equations35, 200-231 (1980) · Zbl 0415.35018 · doi:10.1016/0022-0396(80)90040-6
[4] Ang, D.D., Dinh, A.P.N.: On the strongly damped wave equation:u tt ??u??u t +f(u)=0. SIAM J. Math. Anal.19, 1409-1418 (1988) · Zbl 0685.35071 · doi:10.1137/0519103
[5] Aviles, P., Sandefur, J.: Nonlinear second order equations with applications to partial differential equations. J. Differ. Equations58, 404-427 (1985) · Zbl 0572.34004 · doi:10.1016/0022-0396(85)90008-7
[6] Christodoulou, D.: Global solutions of nonlinear hyperbolic equations for small initial data. Commun. Pure Appl. Math.39, 267-282 (1986) · Zbl 0612.35090 · doi:10.1002/cpa.3160390205
[7] Dafermos, C.M.: The mixed initial-boundary value problem for the equations of non-linear one-dimensional visco-elasticity. J. Differ. Equations6, 71-86 (1969) · Zbl 0218.73054 · doi:10.1016/0022-0396(69)90118-1
[8] Engler, H.: Strong solutions for strongly damped quasilinear wave equations. Contemp. Math.64, 219-237 (1987) · Zbl 0638.35054
[9] Friedman, A., Necas, J.: Systems of nonlinear wave equations with nonlinear viscosity. Pac. J. Math.135, 29-55 (1988) · Zbl 0685.35070
[10] Greenberg, J.M.: On the existence, uniqueness, and stability of the equation ?0 X tt =E(X x )X xx +X xxt . J. Math. Anal. Appl.25, 575-591 (1969) · Zbl 0192.44803 · doi:10.1016/0022-247X(69)90257-1
[11] Greenberg, J.M., MacCamy, R.C., Mizel, J.J.: On the existence, uniqueness, and stability of the equation On the existence, uniqueness, and stability of the equation ?’(u x )u xx ??u xxt =?0 u tt . J. Math. Mech.17, 707-728 (1968) · Zbl 0157.41003
[12] John, F.: Formation of singularities in one-dimensional nonlinear wave propagation. Commun. Pure Appl. Math.27, 370-405 (1974) · Zbl 0302.35064 · doi:10.1002/cpa.3160270307
[13] Kawashima, S., Shibata, Y.: Global existence and exponential stability of small solutions to nonlinear viscoelasticity. Commun. Math. Phys. (to appear) · Zbl 0779.35066
[14] Klainerman, S.: Global existence for nonlinear wave equations. Commun. Pure Appl. Math.33, 43-101 (1980) · Zbl 0414.35054 · doi:10.1002/cpa.3160330104
[15] Klainerman, S.: Long-time behavior of solutions to nonlinear evolution equations. Arch. Ration. Mech. Anal.78, 73-98 (1982) · Zbl 0502.35015 · doi:10.1007/BF00253225
[16] Klainerman, S.: Uniform decay estimates and the Lorentz invariance of the classical wave equation. Commun. Pure Appl. Math.38, 321-332 (1985) · Zbl 0635.35059 · doi:10.1002/cpa.3160380305
[17] Klainerman, S.: The null condition and global existence to nonlinear wave equations. In: Nicolaenko, B. et al. (eds.) Nonlinear systems of partial differential equations in applied mathematics. (Lect. Appl. Math., vol. 23, pp. 293-326) Providence, RI: Am. Math. Soc. 1986
[18] Klainerman, S., Majda, A.: Formation of singularities for wave equations including the nonlinear vibrating string. Commun. Pure Appl. Math.33, 241-263 (1980) · Zbl 0443.35040 · doi:10.1002/cpa.3160330304
[19] Klainerman, S., Ponce, G.: Global small amplitude solutions to nonlinear evolution equations. Commun. Pure Appl. Math.36, 133-141 (1983) · Zbl 0509.35009 · doi:10.1002/cpa.3160360106
[20] Lax, P.D.: Development of singularities of solutions of nonlinear hyperbolic partial differential equations. J. Math. Phys.5, 611-613 (1964) · Zbl 0135.15101 · doi:10.1063/1.1704154
[21] Liu, T.P.: Development of singularities in the nonlinear waves for quasi-linear hyperbolic partial differential equations. J. Differ. Equations33, 92-111 (1979) · Zbl 0404.35071 · doi:10.1016/0022-0396(79)90082-2
[22] MacCamy, R.C., Mizel, V.J.: Existence and nonexistence in the large of solutions of quasilinear wave equations. Arch. Ration. Mech. Anal.25, 299-320 (1967) · Zbl 0146.33801 · doi:10.1007/BF00250932
[23] Matsumura, A.: Global existence and asymptotics of the solutions of the second-order quasilinear hyperbolic equations with first order dissipation. Publ. Res. Inst. Math. Sci. Ser.A13, 349-379 (1977) · Zbl 0371.35030 · doi:10.2977/prims/1195189813
[24] Matsumura, A.: Initial value problems for some quasilinear partial differential equations in mathematical physics. Thesis, Kyoto University 1980
[25] Matsumura, A., Nishida, T.: The initial value problem for the equations of motion of viscous and heat-conductive gases. J. Math. Kyoto Univ.20, 67-104 (1980) · Zbl 0429.76040
[26] Matsumura, A., Nishida, T.: Initial boundary value problems for the equation of motion of compressible viscous and heat-conductive fluids. Commun. Math. Phys.89, 445-464 (1983) · Zbl 0543.76099 · doi:10.1007/BF01214738
[27] Nishida, T.: Nonlinear hyperbolic equations and related topics in fluid dynamics, Publ. Math. Orsay (1978) · Zbl 0392.76065
[28] Pecher, H.: On global regular solutions of third order partial differential equations. J. Math. Anal. Appl.73, 278-299 (1980) · Zbl 0429.35057 · doi:10.1016/0022-247X(80)90033-5
[29] Ponce, G.: Global existence of small solutions to a class of nonlinear evolution equation. Nonlinear Anal., Theory Methods Appl.9, 399-418 (1985) · Zbl 0576.35023 · doi:10.1016/0362-546X(85)90001-X
[30] Racke, R.: Lectures on nonlinear evolution equations. Braunschweig Wiesbaden: Vieweg 1992 · Zbl 0811.35002
[31] Racke, R., Shibata, Y.: Global smooth solutions and asymptotic stability in one-dimensional nonlinear thermoelasticity. Arch. Ratio. Mech. Anal.116, 1-34 (1991) · Zbl 0756.73012 · doi:10.1007/BF00375601
[32] Shatah, J.: Global existence of small solutions to nonlinear evolution equations. J. Differ. Equations46, 409-425 (1982) · Zbl 0518.35046 · doi:10.1016/0022-0396(82)90102-4
[33] Shibata, Y.: On the global existence of classical solutions of second order fully nonlinear hyperbolic equations with first order dissipation in the exterior domain. Tsukuba J. Math.7, 1-68 (1983) · Zbl 0524.35071
[34] Shibata, Y., Tsutsumi, Y.: On a global existence theorem of small amplitude solutions for nonlinear wave equations in an exterior domain. Math. Z.191, 165-199 (1986) · Zbl 0592.35028 · doi:10.1007/BF01164023
[35] Shibata, Y., Zheng, S.: On some nonlinear hyperbolic systems with damping boundary condition. Nonlinear Anal., Theory Methods Appl.17, 233-266 (1991) · Zbl 0772.35031 · doi:10.1016/0362-546X(91)90050-B
[36] Slemrod, M.: Global existence, uniqueness, and asymptotic stability of classical smooth solutions in the one-dimensional non-linear thermoelasticity. Arch. Ration. Mech. Anal.76, 97-133 (1981) · Zbl 0481.73009 · doi:10.1007/BF00251248
[37] Strauss, W.: The energy method in nonlinear partial differential equations. (Notas Mat., no. 47) Riode Janeiro: IMPA 1969 · Zbl 0233.35001
[38] Tanabe, H.: Equations of evolution. (Monogr. Stud. Math. London San Francisco Melbourne: Pitman 1979
[39] Webb, G.F.: Existence and asymptotic behavior for a strongly damped nonlinear wave equation. Can. J. Math.32, 631-643 (1980) · Zbl 0432.35046 · doi:10.4153/CJM-1980-049-5
[40] Yamada, Y.: Some remarks on the equationy tt ??(y x )y xx ?y xtx =f. Osaka J. Math.17, 303-323 (1980) · Zbl 0446.35071
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