For the existence of weak solutions \[ \frac{\partial } {{\partial t}}\left( {|u|^{\beta - 1} u} \right) = {\operatorname{div}}\left( {|\nabla u|^{p 2} \nabla \right)\quad {\text{with}}\,|u|^{\beta u( \cdot ,0) \mu ( ), \] we give a sufficient condition for growth order initial data $\mu (x)$ as $|x| \to \infty $.

We studythe existence and the large time behavior of global-in-time solutions a nonlinear integral equation with generalized heat kernel\begin{eqnarray*} & u(x,t)=\int_{{\mathbb R}^N}G(x-y,t)\varphi(y)dy\\ \qquad\quad+\int_0^t\int_{{\mathbb R}^N}G(x-y,t-s)F(y,s,u(y,s),\dots,\nabla^\ell u(y,s))dyds,\end{eqnarray*}where $\varphi\in W^{\ell,\infty}({\mathbb R}^N)$ $\ell\in\{0,1,\dots\}$.The arguments this paper are applicable tothe Cauchy problem for various parabolic equationssuch as...

We establish the local existence and uniqueness of solutionsof heat equation with a nonlinear boundary conditionfor initial data in uniformly $L^r$ spaces.Furthermore, we study sharp lower estimates blow-up timeof solutions $\lambda\psi$ as $\lambda\to 0$ or $\lambda\to\infty$and solutions.

We study the large time behavior of solutions for Cauchy problem, ∂ t u = Δu + a(x, t)u in ℝ N × (0, ∞), (x, 0) Q(x) , where Q ∈ L 1 (ℝ (1+|x| K ) dx) with ≥ 0 and ∥a(t)∥ ∞ O(t -A as → some A > 1. In this paper we classify decay rate give precise estimates on difference between their asymptotic profiles. Furthermore, an application, discuss global semilinear heat equation, λ|u| p-1 u, λ p

We study the large time behavior of positive solutions forthe Laplace equation on half-space with a nonlinear dynamical boundary condition.We show convergence to Poisson kernel in suitable sense provided initial dataare sufficiently small.

Abstract We investigate some geometric properties of level sets the solutions parabolic problems in convex rings. introduce notion quasi‐concavity , which involves time and space jointly is a stronger property than spatial quasi‐concavity, study convexity superlevel (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

We introduce the notion of $\alpha$-parabolic quasi-concavity for functions space and time, which extends usual parabolic quasi-cocavity introduced in [18]. Then we investigate solutions to problems with vanishing initial datum. The results here obtained are generalizations some

Let u be a type I blowing up solution of the Cauchy–Dirichlet problem for semilinear heat equation, \left\{\begin{matrix} \partial _{t}u = \mathrm{\Delta }u + u^{p}, & x \in \Omega ,\:t > 0, \\ u(x,t) u(x,0) \varphi (x), , \end{matrix}\right. where is (possibly unbounded) domain in \mathbf{R}^{N} N \geq 1 and p . We prove that, if L^{\infty }(\Omega ) \cap L^{q}(\Omega some q [1,\infty) then blow-up set bounded. Furthermore, we give sufficient condition solutions not to blow on...

We consider the Schrödinger operator H = − Δ + V ( | x ) with radial potential which may have singularity at 0 and a quadratic decay infinity. First, we study structure of positive harmonic functions give their precise behavior. Second, under quite general conditions prove an upper bound for correspond heat kernel p , y t type < ⩽ C N / 2 U min { } exp all x, ∈ R > where is function H. Third, if A weight on then lower similar type.

In this paper we obtain necessary conditions and sufficient for the solvability of problem $({\rm P})\, \{ \partial_t u=\Delta u,\, x\in{\bf R}^N_+,\,t>0; \partial_\nu u=u^p,\, x\in\partial{\bf R}^N_+,\, t>0; u(x,0)=\mu(x)\ge 0,\, x\in D:=\overline{{\bf R}^N_+} \}$, where $N\ge 1$, $p>1$, $\mu$ is a nonnegative measurable function in ${\bf R}^N_+$ or Radon measure R}^N$ with $\mbox{supp}\,\mu\subset D$. Our enable us to identify strongest singularity initial data P})$.

We show the existence and uniqueness of initial traces nonnegative solutions to a semilinear heat equation on half space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript N"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>N</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {R}^N</mml:annotation>...

Abstract In this paper we consider degenerate parabolic equations, and obtain an interior a boundary Harnack inequalities for nonnegative solutions to the equations. Furthermore boundedness continuity of solutions.