Large-time behaviour of solutions to non-linear wave equations: higher-order asymptotics

Abstract

Systems of wave equations may fail to be globally well posed, even for small initial data. Attempts to classify systems into well and ill-posed categories work by identifying structural properties of the equations that can work as indicators of well-posedness. The most famous of these are the null and weak null conditions. As noted by Keir, related formulations may fail to properly capture the effect of undifferentiated terms in systems of wave equations. We show that this is because null conditions are good for categorising behaviour close to null infinity, but not at timelike infinity. In this paper, we propose an alternative condition for semilinear equations that work for undifferentiated non-linearities as well. We illustrate the strength of this new condition by proving global well and ill-posedness statements for some systems of equation that are not critical according to the our classification. Furthermore, we given two examples of systems satisfying the weak null condition with global ill-posedness due to undifferentiated terms, thereby disproving the weak null conjecture as stated in [DP18]. Contents 1. Introduction 1 2. The role of asymptotics 12 3. Finding critical exponents 20 4. Technical lemmas for decay 28 5. Testing the hunch: global existence 34 6. Blow-up results 46 References 54

Asymptotics for conservation laws involving Lévy diffusion generators

“…1.2] applies in this case with no essential changes, we will not give the details of calculations (cf. [28] and [17]). …”

“…As a result, the assumptions on the operators L could be relaxed. The tools permitting us to deal with nonlinear terms are reminiscent of those employed in papers [14], [28], [17], [5]. However, the use of a mixture of tools related to semigroup theory and the logarithmic Sobolev inequalities seems to be novel in the context of such nonlinear equations.…”

Ott–Sudan–Ostrovskiy type equations on a segment with large initial data

“…7 648 E. Kaikina ZAMP Here and below p α is the main branch of the complex analytic function in the half-complex plane ℜp ≥ 0, so that 1 α = 1 .We make a cut along a contour Γ Γ = z ∈ C, z ∈ ∞e i(−2π+β) , 0e i(−2π+β) ∪ 0e iβ , ∞e iβ , (1.2) that is we choose arg z ∈ [−2π + β, β) for any complex z ∈ C, β ∈ π 2 , 3π 2 . Recently, much attention has been drawn to the study of the global existence and large-time asymptotic behavior of solutions to the Cauchy problem for nonlinear evolution equations (see,for example, [2]- [10], [12], [13], [15]- [18], [21], [22], [27], [28], [30]- [39]). Nonlinear pseudodifferential equations on a half-line were studied in the book [20].…”

Long-time behavior for a nonlocal convection diffusion equation

“…In this case the decay of solutions has been studied under various assumptions on the regularity of the solutions, i.e. u(0) ∈ L 1 (R) ∩ H 2 (R) in [2], small L 1 (R) ∩ H 1 (R) solutions in [12]. A different approach with a better convergence to the asymptotic profile has been obtained in [14].…”

“…A different approach with a better convergence to the asymptotic profile has been obtained in [14]. However, the asymptotic profile in [14] is not explicit as the one obtained in [12] or [5].…”