Blow-Up Phenomena and Asymptotic Profiles Passing from H 1-Critical to Super-Critical Quasilinear Schrödinger Equations

We study the asymptotic profile, as ℏ → 0 {\hbar\rightarrow 0} , of positive solutions to - ℏ 2 ⁢ Δ ⁢ u + V ⁢ ( x ) ⁢ u - ℏ 2 + γ ⁢ u ⁢ Δ ⁢ u 2 = K ⁢ ( x ) ⁢ | u | p - 2 ⁢ u , x ∈ ℝ N , -\hbar^{2}\Delta u+V(x)u-\hbar^{2+\gamma}u\Delta u^{2}=K(x)\lvert u\rvert^{p-2% }u,\quad x\in\mathbb{R}^{N}, where γ ⩾ 0 {\gamma\geqslant 0} is a parameter with relevant physical interpretations, V and K are given potentials and the dimension N is greater than or equal to 5, as we look for finite L 2 {L^{2}} -energy solutions. We investigate the concentrating behavior of solutions when γ > 0 {\gamma>0} and, differently from the case γ = 0 {\gamma=0} where the leading potential is V, the concentration is here localized by the source potential K. Moreover, surprisingly for γ > 0 {\gamma>0} we find a different concentration behavior of solutions in the case p = 2 ⁢ N N - 2 {p=\frac{2N}{N-2}} and when 2 ⁢ N N - 2 < p < 4 ⁢ N N - 2 {\frac{2N}{N-2}<p<\frac{4N}{N-2}} . This phenomenon does not occur when γ = 0 {\gamma=0} .

Diffusion-induced blowup solutions for the shadow limit model of a singular Gierer–Meinhardt system

In this paper, we provide a thorough investigation of the blowing up behavior induced via diffusion of the solution of the following non-local problem: [Formula: see text] where [Formula: see text] is a bounded domain in [Formula: see text] with smooth boundary [Formula: see text] such problem is derived as the shadow limit of a singular Gierer–Meinhardt system, Kavallaris and Suzuki [On the dynamics of a non-local parabolic equation arising from the Gierer–Meinhardt system, Nonlinearity (2017) 1734–1761; Non-Local Partial Differential Equations for Engineering and Biology: Mathematical Modeling and Analysis, Mathematics for Industry, Vol. 31 (Springer, 2018)]. Under the Turing type condition [Formula: see text] we construct a solution which blows up in finite time and only at an interior point [Formula: see text] of [Formula: see text] i.e. [Formula: see text] where [Formula: see text] More precisely, we also give a description on the final asymptotic profile at the blowup point [Formula: see text] and thus we unveil the form of the Turing patterns occurring in that case due to driven-diffusion instability. The applied technique for the construction of the preceding blowing up solution mainly relies on the approach developed in [F. Merle and H. Zaag, Reconnection of vortex with the boundary and finite time quenching, Nonlinearity 10 (1997) 1497–1550] and [G. K. Duong and H. Zaag, Profile of a touch-down solution to a nonlocal MEMS model, Math. Models Methods Appl. Sci. 29 (2019) 1279–1348].

Dissipative structure and asymptotic profiles for symmetric hyperbolic systems with memory

We study symmetric hyperbolic systems with memory-type dissipation and investigate their dissipative structures under Craftsmanship condition. We treat two cases: memory-type diffusion and memory-type relaxation, and observe that the dissipative structures of these two cases are essentially different. Namely, we show that the dissipative structure of the system with memory-type diffusion is of the standard type, while that of the system with memory-type relaxation is of the regularity-loss type. Moreover, we investigate the asymptotic profiles of the solutions for [Formula: see text]. In the diffusion case, it is proved that the systems with memory and without memory have the same asymptotic profile for [Formula: see text], which is given by the superposition of linear diffusion waves. We have the same result also in the relaxation case under enough regularity assumption on the initial data.

Corrigendum to: The asymptotic profile of an eta-theta quotient related to entanglement entropy in string theory

This corrigendum serves to correct the article [1, Theorem 1.1]. In doing so, we correct the proof and statement of Theorem 3.7, and see that one may disregard Proposition 3.4, Lemma 3.5 and Proposition 3.6.

A probabilistic view on the long-time behaviour of growth-fragmentation semigroups with bounded fragmentation rates

The growth-fragmentation equation models systems of particles that grow and reproduce as time passes. An important question concerns the asymptotic behaviour of its solutions. Bertoin and Watson ( $2018$ ) developed a probabilistic approach relying on the Feynman-Kac formula, that enabled them to answer to this question for sublinear growth rates. This assumption on the growth ensures that microscopic particles remain microscopic. In this work, we go further in the analysis, assuming bounded fragmentations and allowing arbitrarily small particles to reach macroscopic mass in finite time. We establish necessary and sufficient conditions on the coefficients of the equation that ensure Malthusian behaviour with exponential speed of convergence to the asymptotic profile. Furthermore, we provide an explicit expression of the latter.

Asymptotic Profile for Diffusion Wave Terms of the Compressible Navier–Stokes–Korteweg System

The asymptotic profile for diffusion wave terms of solutions to the compressible Navier–Stokes–Korteweg system is studied on R2. The diffusion wave with time-decay estimate was studied by Hoff and Zumbrun (1995, 1997), Kobayashi and Shibata (2002), and Kobayashi and Tsuda (2018) for compressible Navier–Stokes and compressible Navier–Stokes–Korteweg systems. Our main assertion in this paper is that, for some initial conditions given by the Hardy space, asymptotic behaviors in space–time L2 of the diffusion wave parts are essentially different between density and the potential flow part of the momentum. Even though measuring by L2 on space, decay of the potential flow part is slower than that of the Stokes flow part of the momentum. The proof is based on a modified version of Morawetz’s energy estimate, and the Fefferman–Stein inequality on the duality between the Hardy space and functions of bounded mean oscillation.