Strong solutions for semilinear wave equations with damping and source terms

Abstract

This article deals with local existence of strong solutions for semilinear wave equations with power-like interior damping and source terms. A long-standing restriction on the range of exponents for the two nonlinearities governs the literature on wellposedness of weak solutions of finite energy. We show that this restriction may be eliminated for the existence of higher regularity solutions by employing natural methods that use the physics of the problem. This approach applies to the Cauchy problem posed on the entire ℝ ⁿ as well as for initial boundary problems with homogeneous Dirichlet boundary conditions.

Keywords:

• wave equation
• local existence
• finite speed of propagation
• nonlinear damping
• interior source
• strong solutions

AMS Subject Classifications::

• Primary: 35L05
• Secondary: 35L20

Notes

Notes

1. A ‘classical’ potential well is generated by a function , for some constants A ₀ > 0, q > 2. This is a C ¹ function and it has one local maximum at x = (A ₀ q)^−1/q−1, whereas the above function φ₁ may not be differentiable at the local maximum, or it may have two ‘humps’. Nevertheless, as we show in the sequel the arguments follow the same way for this slightly modified function.

2. Note that all these operations are legitimate due to our higher regularity assumption (a4) from section 4.1: