Well-posedness, global existence and large time behavior for Hardy–Hénon parabolic equations

Optimal well-posedness and forward self-similar solution for the Hardy-Hénon parabolic equation in critical weighted Lebesgue spaces

“…It suffices to prove the inequality for e ∆ f and then resort to a dilation argument as in the proof of [1,Proposition 2.1].…”

“…In [22], unconditional uniqueness has been established for the Hardy case γ < 0. Concerning earlier conditional uniqueness when γ < 0, we refer to [1,2]. Lastly, we refer to [15] for the analysis of the problem (1.1) with an external forcing term in addition to the nonlinear term.…”

“…Later, the case α F (d, γ) < α < 1 + 2(2+γ) d−2 is treated in [9] under some additional restriction on γ, namely γ ≤ 0 for d ≥ 4 and γ ≤ √ 3 − 1 for d = 3. In [1,Theorem 1.4], the existence of self-similar solutions that are not necessarily radially symmetric has been proved for all α > α F (d, γ), but only for the Hardy case γ < 0 (See also [3]). Our result (Theorem 1.9) covers all the previous results and asserts the existence of non-radial forward self-similar solutions for γ and α such that − min(2, d) < γ and α > α F (d, γ).…”

Local well‐posedness and blow‐up for an inhomogeneous nonlinear heat equation

“…We quote Pinsky, where a Fujita exponent is given and the behavior of the life span of solutions was studied under some assumptions on the nonlinearity and the weight . Note the case of bounded domain was investigated in Loayza, and the case of singular weight (ie, ) was studied in Slimene et al…”

Initial-boundary value problem for a fourth-order plate equation with Hardy-Hénon potential and polynomial nonlinearity

“…For example, the Efimov states (the circumstances that the two-particle attraction is so weak that any two bosons can not form a pair, but the three bosons can be stable bound states): see e.g., [16]; effects on dipole-bound anions in polar molecules: see e.g., [4,7,28]; capture of matter by black holes (via near-horizon limits): see e.g., [9,19]; the motions of cold neutral atoms interacting with thin charged wires (falling in the singularity or scattering): see e.g., [5,12]; the renormalization group of limit cycle in nonrelativistic quantum mechanics: see e.g., [6,8]; and so on. The are a lot of studies of evolution equations with this type of potentials, see, for example, [1,2,3,45,32,50] for parabolic equation, [11,49,52,39] for wave equations, and [24,22,23,42,47,48,53] for Schrödinger equation. However, as far as we know, there seems little studies of fourth-order plate equation with Hardy-Hénon potentials.…”