Construction and stability of blowup solutions for a non-variational semilinear parabolic system

Abstract

+∞] the maximal existence time of the classical solution (u, v) of problem (1.1). If T < +∞, then the solution blows up in finite time T in the sense thatlim t→T ( u(t) L ∞ (R N ) + v(t) L ∞ (R N ) ) = +∞.

Construction and stability of type I blowup solutions for non-variational semilinear parabolic systems

“…As a consequence of our technique, the constructed solutions are stable under a small perturbation of initial data. The results and the main arguments presented in this note can be found in our papers [19,20].…”

“…In this note we exhibit Type I blowup solutions for system (1.1) and give the first complete description of its asymptotic behavior. More precisely, we prove in [19] the following theorem. Theorem 1.1 (Type I blowup solutions for (1.1)-(1.2) and its asymptotic behavior, [19]).…”

“…In [19,20], we consider the semilinear parabolic system    ∂ t u = ∆u + f (v), 1) with N ≥ 1, µ > 0 and u(0), v(0) = u 0 , v 0 , where (u, v)(t) : x ∈ R N → R 2 and the nonlinearity has no gradient structure taking of the particular form f (v) = v|v| p−1 and g(u) = u|u| q−1 with p, q > 1, (1.2) or f (v) = e pv and g(u) = e qu with p, q > 0. (1.3) System (1.1) represents a simple model of a reaction-diffusion system describing heat propagation in a two-component combustible mixture and, as such, it has been the subject of intensive investigation from the last two decades (see [44], [50] and references therein).…”

Prescribing the center of mass of a multi-soliton solution for a perturbed semilinear wave equation

“…-In Part 1, in order to prove (31) and (32), we project equation (69) using the projector Π d * i (s) l defined in (20) with l = 0, 1 and d * i (s) = d i (s) 1+ν i (s) . -In Part 2, we will find a Lyapunov functional for equation (69), which is equivalent to the norm squared, and we thus obtain estimate (33).…”

“…. , k be fixed and l = 0 or 1, the projector Π d * i (s) l defined in (20) is now applied for each term of equation (69). Thanks to (29) together with the analysis of [51, Appendix C], [52, Appendix C] and [12, Appendix A], we easily obtain the following estimates related to the terms not involving f and g:…”