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Register a new userWell-posedness for Hardy-Henon parabolic equations with fractional Brownian noise
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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 $q$.
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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 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.
We consider time fractional parabolic equations in both divergence and non-divergence form when the leading coefficients $a^{ij}$ are measurable functions of $(t,x_1)$ except for $a^{11}$ which is a measurable function of either $t$ or $x_1$. We obtain the solvability in Sobolev spaces of the equations in the whole space, on a half space, or on a partially bounded domain. The proofs use a level set argument, a scaling argument, and embeddings in fractional parabolic Sobolev spaces for which we give a direct and elementary proof.
In this paper, we establish several Liouville type theorems for entire solutions to fractional parabolic equations. We first obtain the key ingredients needed in the proof of Liouville theorems, such as narrow region principles and maximum principles for antisymmetric functions in unbounded domains, in which we remarkably weaken the usual decay condition $u to 0$ at infinity with respect to the spacial variables to a polynomial growth on $u$ by constructing auxiliary functions.Then we derive monotonicity for the solutions in a half space $mathbb{R}_+^n times mathbb{R}$ and obtain some new connections between the nonexistence of solutions in a half space $mathbb{R}_+^n times mathbb{R}$ and in the whole space $mathbb{R}^{n-1} times mathbb{R}$ and therefore prove the corresponding Liouville type theorems. To overcome the difficulty caused by the non-locality of the fractional Laplacian, we introduce several new ideas which will become useful tools in investigating qualitative properties of solutions for a variety of non-local parabolic problems.
We develop a strategy making extensive use of tent spaces to study parabolic equa-tions with quadratic nonlinearities as for the Navier-Stokes system. We begin with a new proof of the well-known result of Koch and Tataru on the well-posedness of Navier-Stokes equations in R^n with small initial data in BMO^{-1}(R^n). We then study another model where neither pointwise kernel bounds nor self-adjointness are available.
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