Existence and Asymptotic Behavior for a Strongly Damped Nonlinear Wave Equation

wt(x, 0) = $(x), x £ 12 w(s, 0 = 0, x Ç 612, t ^ 0. In (1.1) 12 is a smooth bounded domain in R, w = 1, 2, 3, a > 0, and / G C 2 (R;R) with/ ' (x) ^ £o for all x Ç R (where c0 is a nonnegative constant), lim §up\x\^+mf{x)/x S 0, and /(0) = 0. Our objective will be to establish the existence of unique strong global solutions to (1.1) and investigate their behavior as t —> +oo . Our approach takes advantage of the semilinear character of (1.1) and reformulates the problem as an abstract ordinary differential equation in a Banach space. We identify the Laplacian A in 12 with the infinitesimal generator of a strongly continuous semigroup of operators in L(12) and we define F: D(A)->L(Q) by F(4>)(x) = / («(*)) . The problem (1.1) may then be written abstractly as

[1]  Thomas K. Caughey,et al.  Existence, uniqueness, and stability of solutions of a class of nonlinear partial differential equations , 1975 .

[2]  Constantine M. Dafermos,et al.  The Riemann problem for certain classes of hyperbolic systems of conservation laws , 1976 .

[3]  Initial value problems and time-periodic solutions for a nonlinear wave equation† , 1957 .

[4]  Mitsuhiro Nakao.  CONVERGENCE OF SOLUTIONS OF THE WAVE EQUATION WITH A NONLINEAR DISSIPATIVE TERM TO THE STEADY STATE , 1976 .

[5]  I. Segal.  Non-Linear Semi-Groups , 1963 .

[6]  Hideo Kawarada,et al.  On Solutions of Nonlinear Wave Equations , 1971 .

[7]  Chung-lie Wang.  A Uniqueness Theorem on the Degenerate Cauchy Problem(1) , 1975, Canadian Mathematical Bulletin.

[8]  P. Brenner,et al.  Global classical solutions of nonlinear wave equations , 1981 .

[9]  J. Ball.  On the asymptotic behavior of generalized processes, with applications to nonlinear evolution equations , 1978 .

[10]  J. Greenberg.  Smooth and time-periodic solutions to the quasilinear wave equation , 1975 .

[11]  G. Webb.  A Bifurcation Problem for a Nonlinear Hyperbolic Partial Differential Equation , 1979 .

[12]  P. Davis.  A quasilinear hyperbolic and related third-order equations , 1975 .

[13]  V. Mizel,et al.  Existence and nonexistence in the large of solutions of quasilinear wave equations , 1967 .

[14]  A. Pazy,et al.  A class of semi-linear equations of evolution , 1975 .

[15]  J. Clements,et al.  EXISTENCE THEOREMS FOR A QUASILINEAR EVOLUTION EQUATION , 1974 .