1,715 research outputs found
Solvability of the heat equation with a nonlinear boundary condition
We obtain necessary conditions and sufficient conditions for the solvability
of the heat equation in a half-space of with a nonlinear boundary
condition. Furthermore, we study the relationship between the life span of the
solution and the behavior of the initial function
Heat equation with a nonlinear boundary condition and uniformly local spaces
We establish the local existence and the uniqueness of solutions of the heat
equation with a nonlinear boundary condition for the initial data in uniformly
local spaces. Furthermore, we study the sharp lower estimates of the
blow-up time of the solutions with the initial data as
or and the lower blow-up estimates of the
solutions
Parabolic power concavity and parabolic boundary value problems
This paper is concerned with power concavity properties of the solution to
the parabolic boundary value problem \begin{equation} \tag{}
\left\{\begin{array}{ll} \partial_t u=\Delta u +f(x,t,u,\nabla u) &
\mbox{in}\quad\Omega\times(0,\infty),\vspace{3pt}\\ u(x,t)=0 &
\mbox{on}\quad\partial \Omega\times(0,\infty),\vspace{3pt}\\ u(x,0)=0 &
\mbox{in}\quad\Omega, \end{array} \right. \end{equation} where is a
bounded convex domain in and is a nonnegative continuous
function in . We give a
sufficient condition for the solution of to be parabolically power
concave in
Large time behavior of ODE type solutions to nonlinear diffusion equations
Consider the Cauchy problem for a nonlinear diffusion equation
\begin{equation} \tag{P} \left\{ \begin{array}{ll} \partial_t u=\Delta
u^m+u^\alpha & \quad\mbox{in}\quad{\bf R}^N\times(0,\infty),\\
u(x,0)=\lambda+\varphi(x)>0 & \quad\mbox{in}\quad{\bf R}^N, \end{array} \right.
\end{equation} where , , and with and . Then the positive solution to problem (P) behaves
like a positive solution to ODE in and it
tends to as . In this paper we obtain the precise
description of the large time behavior of the solution and reveal the
relationship between the behavior of the solution and the diffusion effect the
nonlinear diffusion equation has
Sharp decay estimates in Lorentz spaces for nonnegative Schr\"odinger heat semigroups
Let be a nonnegative Schr\"odinger operator on , where and is a radially symmetric function decaying
quadratically at the space infinity. In this paper we consider the
Schr\"odinger heat semigroup , and make a complete table of the decay
rates of the operator norms of in the Lorentz spaces as
Parabolic Minkowski convolutions of viscosity solutions to fully nonlinear equations
This paper is concerned with the Minkowski convolution of viscosity solutions
of fully nonlinear parabolic equations. We adopt this convolution to compare
viscosity solutions of initial-boundary value problems in different domains. As
a consequence, we can for instance obtain parabolic power concavity of
solutions to a general class of parabolic equations. Our results apply to the
Pucci operator, the normalized -Laplacians with , the Finsler
Laplacian and more general quasilinear operators
Asymptotics for a nonlinear integral equation with a generalized heat kernel
This paper is concerned with a nonlinear integral equation where , for some . Here is a
generalization of the heat kernel. We are interested in the asymptotic
expansions of the solution of behaving like a multiple of the integral
kernel as
A supercritical scalar field equation with a forcing term
This paper is concerned with the elliptic problem for a scalar field equation
with a forcing term \begin{equation} \tag{P}-\Delta u+u=u^p+ \kappa \mu \quad
\mbox{in} \quad{\bf R}^N, \quad u>0 \quad \mbox{in} \quad {\bf R}^N, \quad
u(x)\to 0\quad \mbox{as} \quad |x| \to \infty, \end{equation} where ,
, and is a Radon measure in with a compact
support. Under a suitable integrability condition on , we give a complete
classification of the solvability of problem~(P) with . Here
is the Joseph-Lundgren exponent defined by
p_{JL} :=\infty\quad\mbox{if}\quad N\le 10,
\qquad p_{JL}:=\frac{(N-2)^2-4N+8\sqrt{N-1}}{(N-2)(N-10)}\quad \text{if}
\quad N\ge 11. Comment: 29 page
New characterizations of log-concavity
We introduce a notion of -concavity which largely generalizes the usual
concavity. By the use of the notions of closedness under positive scalar
multiplication and closedness under positive exponentiation we characterize
power concavity and power log-concavity among nontrivial -concavities,
respectively. In particular, we have a characterization of log-concavity as the
only -concavity which is closed both under positive scalar multiplication
and positive exponentiation. Furthermore, we discuss the strongest
-concavity preserved by the Dirichlet heat flow, characterizing
log-concavity also in this connection.Comment: 21pages. Comments are welcome
To logconcavity and beyond
In 1976 Brascamp and Lieb proved that the heat flow preserves logconcavity.
In this paper, introducing a variation of concavity, we show that it preserves
in fact a stronger property than logconcavity and we identify the strongest
concavity preserved by the heat flow.Comment: We removed one result, Theorem 3.2 of the old version, due to a gap
in the proo
- …