Search results for "Blow-up"
showing 9 items of 9 documents
A blow-up result for a nonlinear wave equation on manifolds: the critical case
2021
We consider a inhomogeneous semilinear wave equation on a noncompact complete Riemannian manifold (Formula presented.) of dimension (Formula presented.), without boundary. The reaction exhibits the combined effects of a critical term and of a forcing term. Using a rescaled test function argument together with appropriate estimates, we show that the equation admits no global solution. Moreover, in the special case when (Formula presented.), our result improves the existing literature. Namely, our main result is valid without assuming that the initial values are compactly supported.
Numerical study of blow-up and stability of line solitons for the Novikov-Veselov equation
2017
International audience; We study numerically the evolution of perturbed Korteweg-de Vries solitons and of well localized initial data by the Novikov-Veselov (NV) equation at different levels of the 'energy' parameter E. We show that as |E| -> infinity, NV behaves, as expected, similarly to its formal limit, the Kadomtsev-Petviashvili equation. However at intermediate regimes, i.e. when |E| is not very large, more varied scenarios are possible, in particular, blow-ups are observed. The mechanism of the blow-up is studied.
Stress concentration for closely located inclusions in nonlinear perfect conductivity problems
2019
We study the stress concentration, which is the gradient of the solution, when two smooth inclusions are closely located in a possibly anisotropic medium. The governing equation may be degenerate of $p-$Laplace type, with $1<p \leq N$. We prove optimal $L^\infty$ estimates for the blow-up of the gradient of the solution as the distance between the inclusions tends to zero.
On critical behaviour in generalized Kadomtsev-Petviashvili equations
2016
International audience; An asymptotic description of the formation of dispersive shock waves in solutions to the generalized Kadomtsev–Petviashvili (KP) equation is conjectured. The asymptotic description based on a multiscales expansion is given in terms of a special solution to an ordinary differential equation of the Painlevé I hierarchy. Several examples are discussed numerically to provide strong evidence for the validity of the conjecture. The numerical study of the long time behaviour of these examples indicates persistence of dispersive shock waves in solutions to the (subcritical) KP equations, while in the supercritical KP equations a blow-up occurs after the formation of the disp…
Cyclicity of common slow–fast cycles
2011
Abstract We study the limit cycles of planar slow–fast vector fields, appearing near a given slow–fast cycle, formed by an arbitrary sequence of slow parts and fast parts, and where the slow parts can meet the fast parts in a nilpotent contact point of arbitrary order. Using the notion slow divergence integral, we delimit a large subclass of these slow–fast cycles out of which at most one limit cycle can perturb, and a smaller subclass out of which exactly one limit cycle will perturb. Though the focus lies on common slow–fast cycles, i.e. cycles with only attracting or only repelling slow parts, we present results that are valid for more general slow–fast cycles. We also provide examples o…
High precision numerical approach for Davey–Stewartson II type equations for Schwartz class initial data
2020
We present an efficient high-precision numerical approach for Davey–Stewartson (DS) II type equa- tions, treating initial data from the Schwartz class of smooth, rapidly decreasing functions. As with previous approaches, the presented code uses discrete Fourier transforms for the spatial dependence and Driscoll’s composite Runge–Kutta method for the time dependence. Since DS equations are non-local, nonlinear Schrödinger equations with a singular symbol for the non-locality, standard Fourier methods in practice only reach accuracy of the order of 10−6or less for typical examples. This was previously demonstrated for the defocusing integrable case by comparison with a numerical approach for …
Numerical study of the transverse stability of the Peregrine solution
2020
We generalise a previously published approach based on a multi-domain spectral method on the whole real line in two ways: firstly, a fully explicit 4th order method for the time integration, based on a splitting scheme and an implicit Runge--Kutta method for the linear part, is presented. Secondly, the 1D code is combined with a Fourier spectral method in the transverse variable both for elliptic and hyperbolic NLS equations. As an example we study the transverse stability of the Peregrine solution, an exact solution to the one dimensional nonlinear Schr\"odinger (NLS) equation and thus a $y$-independent solution to the 2D NLS. It is shown that the Peregrine solution is unstable against all…
Gradient estimates for the perfect conductivity problem in anisotropic media
2018
Abstract We study the perfect conductivity problem when two perfectly conducting inclusions are closely located to each other in an anisotropic background medium. We establish optimal upper and lower gradient bounds for the solution in any dimension which characterize the singular behavior of the electric field as the distance between the inclusions goes to zero.
On the critical curve for systems of hyperbolic inequalities in an exterior domain of the half-space
2023
We establish blow-up results for a system of semilinear hyperbolic inequalities in an exterior domain of the half-space. The considered system is investigated under an inhomogeneous Dirichlet-type boundary condition depending on both time and space variables. In certain cases, an optimal criterium of Fujita-type is derived. Our results yield naturally sharp nonexistence criteria for the corresponding stationary wave system and equation.