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Register a new userWell-posedness and global dynamics for the critical Hardy-Sobolev parabolic equation
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We study the Cauchy problem for the semilinear heat equation with the singular potential, called the Hardy-Sobolev parabolic equation, in the energy space. The aim of this paper is to determine a necessary and sufficient condition on initial data below or at the ground state, under which the behavior of solutions is completely dichotomized. More precisely, the solution exists globally in time and its energy decays to zero in time, or it blows up in finite or infinite time. The result on the dichotomy for the corresponding Dirichlet problem is also shown as a by-product via comparison principle.
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The Cauchy problem for the Hardy-Henon parabolic equation is studied in the critical and subcritical regime in weighted Lebesgue spaces on the Euclidean space $mathbb{R}^d$. Well-posedness for singular initial data and existence of non-radial forward self-similar solution of the problem are previously shown only for the Hardy and Fujita cases ($gammale 0$) in earlier works. The weighted spaces enable us to treat the potential $|x|^{gamma}$ as an increase or decrease of the weight, thereby we can prove well-posedness to the problem for all $gamma$ with $-min{2,d}<gamma$ including the Henon case ($gamma>0$). As a byproduct of the well-posedness, the self-similar solutions to the problem are also constructed for all $gamma$ without restrictions. A non-existence result of local solution for supercritical data is also shown. Therefore our critical exponent $s_c$ turns out to be optimal in regards to the solvability.
We study the Hardy-Henon parabolic equations on $mathbb{R}^{N}$ ($N=2, 3$) under the effect of an additive fractional Brownian noise with Hurst parameter $H>maxleft(1/2, N/4right).$ We show local existence and uniqueness of a mid $L^{q}$-solution under suitable assumptions on $q$.
The initial value problem for the $L^{2}$ critical semilinear Schrodinger equation in $R^n, n geq 3$ is considered. We show that the problem is globally well posed in $H^{s}({Bbb R^{n}})$ when $1>s>frac{sqrt{7}-1}{3}$ for $n=3$, and when $1>s> frac{-(n-2)+sqrt{(n-2)^2+8(n-2)}}{4}$ for $n geq 4$. We use the ``$I$-method combined with a local in time Morawetz estimate.
We prove global well-posedness for 3D Dirac equation with a concentrated nonlinearity.
This article is devoted to review the known results on global well-posedness for the Cauchy problem to the Kirchhoff equation and Kirchhoff systems with small data. Similar results will be obtained for the initial-boundary value problems in exterior domains with compact boundary. Also, the known results on large data problems will be reviewed together with open problems.
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