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

Abstract

<jats:title>Abstract</jats:title><jats:p>In this note, we consider the semilinear heat system</jats:p><jats:p><jats:disp-formula id="j_apam-2018-0168_eq_9999_w2aab3b7e4366b1b6b1aab1c14b2aAa"><jats:alternatives><jats:tex-math>\partial_{t}u=\Delta u+f(v),\quad\partial_{t}v=\mu\Delta v+g(u),\quad\mu&gt;0,</jats:tex-math></jats:alternatives></jats:disp-formula></jats:p><jats:p>where the nonlinearity has no gradient structure taking of the particular form</jats:p><jats:p><jats:disp-formula id="j_apam-2018-0168_eq_9998_w2aab3b7e4366b1b6b1aab1c14b4aAa"><jats:alternatives><jats:tex-math>f(v)=v\lvert v\rvert^{p-1}\quad\text{and}\quad g(u)=u\lvert u\rvert^{q-1}\quad% \text{with }p,q&gt;1,</jats:tex-math></jats:alternatives></jats:disp-formula></jats:p><jats:p>or</jats:p><jats:p><jats:disp-formula id="j_apam-2018-0168_eq_9997_w2aab3b7e4366b1b6b1aab1c14b6aAa"><jats:alternatives><jats:tex-math>f(v)=e^{pv}\quad\text{and}\quad g(u)=e^{qu}\quad\text{with }p,q&gt;0.</jats:tex-math></jats:alternatives></jats:disp-formula></jats:p><jats:p>We exhibit type I blowup solutions for this system and give a precise description of its blowup profiles. The method relies on a two-step procedure: the reduction of the problem to a finite-dimensional one via a spectral analysis, and then solving the finite-dimensional problem by a classical topological argument based on index theory. 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 [T.-E. Ghoul, V. T. Nguyen and H. Zaag, Construction and stability of blowup solutions for a non-variational semilinear parabolic system, Ann. Inst. H. Poincaré Anal. Non Linéaire 35 2018, 6, 1577–1630] and [M. A. Herrero and J. J. L. Velázquez, Generic behaviour of one-dimensional blow up patterns, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 19 1992, 3, 381–450].</jats:p>

Refined asymptotics for the blow-up solution of the complex Ginzburg–Landau equation in the subcritical case

“…For parabolic equations, it has been used in [24] and [39] for the complex Ginzburg Landau (CGL) equation with no gradient structure, the critical harmonic heat flow in [31], the two-dimensional Keller-Segel equation in [32] and the nonlinear heat equation involving a nonlinear gradient term in [12,36]. Recently, this method has been applied to various nonvariational parabolic systems in [27] and [13][14][15][16], and to a logarithmically perturbed nonlinear equation in [7][8][9]26]. We also mention a result for a higher-order parabolic equation [17] and in [1,11] two more results for equations involving nonlocal terms.…”

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

“…More generally, constructing a solution to some PDE with a prescribed behavior (not necessarily multi-solitons solutions) is an important question. That question was solved for (gKdV) by Côte [9,10], and also for parabolic equations exhibiting blow-up, like the semilinear heat equation by Bressan [3,4] (with an exponential source), Merle [43], Bricmont and Kupiainen [5], Merle and Zaag in [46,45], Schweyer [57] (in the critical case), Mahmoudi, Nouaili and Zaag [38] (in the periodic case), the complex Ginzburg-Landau equation by Zaag [62], Masmoudi and Zaag in [41] and also Nouaili and Zaag [55], a complex heat equation with no gradient structure by Nouaili and Zaag [54], a gradient perturbed heat equation in the subcritical case by Ebde and Zaag in [15], then by Tayachi and Zaag in the critical case in [58,59] and also by Ghoul, Nguyen and Zaag in [19], a strongly perturbed heat equation in Nguyen and Zaag [53], a non scaling invariant heat equation in Duong, Nguyen, and Zaag [14], two non variational parabolic system Ghoul, Nguyen and Zaag [20,21,22] or a higher order parabolic equation in [23]. Other examples are available for Schrödinger maps (see Merle, Raphaël and Rodnianski [44]), the wave maps (see Ghoul, Ibrahim and Nguyen [17]), and also for the Keller-Segel model (see Raphaël and Schweyer [56], and also Ghoul and Masmoudi [18]).…”