Kazuhiro Ishige

On the existence of solutions of the Cauchy problem for a doubly nonlinear parabolic equation

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

An intrinsic metric approach to uniqueness of the positive Cauchy–Neumann problem for parabolic equations

\mu (x)

Convexity breaking of the free boundary for porous medium equations

as

Asymptotic expansions of solutions of the Cauchy problem for nonlinear parabolic equations

|x| \to \infty

Supersolutions for a class of nonlinear parabolic systems

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The heat kernel of a Schrödinger operator with inverse square potential

Abstract We study uniqueness of nonnegative solutions of the Cauchy–Neumann problem for parabolic equations in unbounded domains, and give a sufficient condition for the uniqueness of nonnegative solutions to hold. We also give a parabolic Harnack inequality with Neumann boundary conditions.

Decay Rate of L q Norms of Critical Schrödinger Heat Semigroups

Is then Pφ(t) convex for every t > 0? (See Section 2 for the definition of α-concavity.) The porous medium equation provides a simple model in many physical situations, in particular, the flow of an isentropic gas through a porous medium; in such a case, u and um−1 represent the density and the pressure of the gas, respectively. Due to its practical interest, regularity and geometric properties of the free boundary ∂Pφ(t) have been extensively studied by many

The gradient theory of the phase transitions in Cahn-Hilliard fluids with Dirichlet boundary conditions

AbstractThis paper is concerned with the Cauchy problem for the nonlinear parabolic equation

Spatial concavity of solutions to parabolic systems

{\partial _t}u| = \vartriangle u + F(x,t,u,\nabla u){\text{ in }}{{\text{R}}^N} \times (0,\infty ),{\text{ }}u(x,0) = \varphi (x){\text{ in }}{{\text{R}}^N},