TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 357, Number 7, Pages 2571–2611 S 0002-9947(05)03880-8 Article electronically published on March 1, 2005 ON NONLINEAR WAVE EQUATIONS WITH DEGENERATE DAMPING AND SOURCE TERMS VIOREL BARBU, IRENA LASIECKA, AND MOHAMMAD A. RAMMAHA Abstract. In this article we focus on the global well-posedness of the differ- ential equation utt - ∆u + |u| k ∂j (ut )= |u| p-1 u in Ω × (0,T ), where ∂j is a sub-differential of a continuous convex function j . Under some conditions on j and the parameters in the equations, we obtain several results on the existence of global solutions, uniqueness, nonexistence and propagation of regularity. Under nominal assumptions on the parameters we establish the existence of global generalized solutions. With further restrictions on the parameters we prove the existence and uniqueness of a global weak solution. In addition, we obtain a result on the nonexistence of global weak solutions to the equation whenever the exponent p is greater than the critical value k + m, and the ini- tial energy is negative. We also address the issue of propagation of regularity. Specifically, under some restriction on the parameters, we prove that solutions that correspond to any regular initial data such that u 0 ∈ H 2 (Ω) ∩ H 1 0 (Ω), u 1 ∈ H 1 0 (Ω) are indeed strong solutions. 1. Introduction 1.1. The model. Let j (s) be a continuous, convex real-valued function defined on R and let ∂j be its sub-differential [3]. Let Ω be a bounded domain in R n with a smooth boundary Γ. This paper is concerned with the solvability of the following initial-boundary value problem: u tt - ∆u + |u| k ∂j (u t )= |u| p-1 u in Ω × (0,T ) ≡ Q T , u(x, 0) = u 0 (x) ∈ H 1 0 (Ω), u t (x, 0) = u 1 (x) ∈ L 2 (Ω), (1.1) u = 0 on Γ × (0,T ), where the problem is studied under the following condition imposed on the convex function j and the parameters k, m, p. Assumption 1.1. • k, m, p ≥ 0. In addition, k ≤ n n-2 , p +1 < 2n n-2 , if n ≥ 3. • Coercivity condition: j (s) ≥ c|s| m+1 , where c> 0. • Strict monotonicity: (∂j (s) - ∂j (v))(s - v) ≥ c 1 |s - v| m+1 , where c 1 > 0. • Continuity: ∂j (s) is single valued and |∂j (s)|≤ c 0 |s| m + c 2 , for some constants c 0 > 0,c 2 ≥ 0. Received by the editors March 31, 2003. 2000 Mathematics Subject Classification. Primary 35L05, 35L20; Secondary 58J45. Key words and phrases. Wave equations, damping and source terms, generalized solutions, weak solutions, blow-up of solutions, sub-differentials, energy estimates. c 2005 American Mathematical Society 2571 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use