Almost global existence for some semilinear wave equations

Abstract

This article establishes the almost global existence of solutions of three-dimensional quadratically semilinear wave equations with the use of only the classical invariance of the equations under translations and spatial rotation. Using these techniques we can handle semilinear wave equations in Minkowski space or semilinear Dirichlet-wave equations in the exterior of a nontrapping obstacle.Our results and approach are related to previous work in the non-obstacle case. In particular, in [2], almost global existence was shown for general, quadratic quasilinear wave equations in Minkowski space, using fully the invariance properties of the wave operator under translations, spatial rotations, scaling, and Lorentz boosts . Subsequently, in [5], the almost global existence of certain divergence-form quadratic quasilinear wave equations was established in Minkowski space, using only the invariance of the wave operator under translations, spatial rotations, and changes of scale. Further techniques and results concerning equations from the theory of elasticity were presented in [9], [10].The main difference between our approach and earlier work is that it only exploits the O(|x| −1 ) decay of solutions of wave equations with compactly supported data, as opposed to the stronger O(t −1 ) decay. Here of course x = (x 1 , x 2 , x 3 ) is the spatial component and t the time component of a space-time vector (t, x) ∈ R + × R 3 . Establishing O(|x| −1 ) decay is considerably easier and can be achieved using only the invariance with respect to translations and spatial rotations. Fortunately, as we shall see, one can set up an iteration argument that only requires this weaker decay to obtain almost global existence. Another advantage to this approach for obstacle problems is that the translation and spatial rotational vector fields essentially preserve the Dirichlet conditions. The homogeneous vector fields used in [2] do not have this property for any obstacle, and consequently it seems difficult to use them for nonlinear obstacle problems.

Nonlinear Hyperbolic Equations in Infinite Homogeneous Waveguides

“…For Theorem 1.3, which concerns existence for semilinear wave equations with Neumann boundary conditions, one just needs to use the special case of (4.3), (4.5) where all of the γ jk vanish identically, i.e., the standard energy estimate for constant coefficients and Neumann conditions. By using this and Proposition 3.6 one obtains Theorem 1.3 using arguments from [8]. The remainder of this section will be dedicated to sketching this proof.…”

“…We remark, though, that when n ≥ 5 it seems that our extension (2.5) of the KSS inequality [8] does give global existence for equations involving bilinear forms in (∂ t u, ∇ x u). We shall study this in a future paper.…”

“…To prove Theorem 1.3 (i.e., Neumann boundary conditions with m = 0), we shall also require weighted L 2 t L 2 x estimates of the type that were first used in [8]. Here and throughout, we will use the notation x = r = 1 + |x| 2 .…”

“…It should be compared to the symmetry condition for multispeed nonlinear hyperbolic systems (see, e.g., [9], [16], [20]). …”

“…However, we can obtain optimal results for certain semilinear equations without too much difficulty using ideas from [8]. Theorem 1.3.…”