On an over-determined problem of free boundary of a degenerate parabolic equation

Jiaqing Pan

Displaying similar documents to “On an over-determined problem of free boundary of a degenerate parabolic equation”

Large time behavior of solutions to a class of doubly nonlinear parabolic equations

Hua Shui Zhan (2008)

Applications of Mathematics

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We study the large time asymptotic behavior of solutions of the doubly degenerate parabolic equation $u_{t} = div (u^{m - 1} {| D u |}^{p - 2} D u) - u^{q}$ with an initial condition $u (x, 0) = u_{0} (x)$ . Here the exponents $m$ , $p$ and $q$ satisfy $m + p \geq 3$ , $p > 1$ and $q > m + p - 2$ .

A global differentiability result for solutions of nonlinear elliptic problems with controlled growths

Luisa Fattorusso (2008)

Czechoslovak Mathematical Journal

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Let $Ω$ be a bounded open subset of $ℝ^{n}$ , $n > 2$ . In $Ω$ we deduce the global differentiability result

$u \in H^{2} (Ω, ℝ^{N})$ for the solutions $u \in H^{1} (Ω, ℝ^{n})$ of the Dirichlet problem

$u - g \in H_{0}^{1} (Ω, ℝ^{N}), - \sum_{i} D_{i} a^{i} (x, u, D u) = B_{0} (x, u, D u)$ with controlled growth and nonlinearity $q = 2$ . The result was obtained by first extending the interior differentiability result near the boundary and then proving the global differentiability result making use of a covering procedure.

Differentiability of weak solutions of nonlinear second order parabolic systems with quadratic growth and nonlinearity $q \geq 2$

Luisa Fattorusso (2004)

Commentationes Mathematicae Universitatis Carolinae

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Let $Ω$ be a bounded open subset of $ℝ^{n}$ , let $X = (x, t)$ be a point of $ℝ^{n} \times ℝ^{N}$ . In the cylinder $Q = Ω \times (- T, 0)$ , $T > 0$ , we deduce the local differentiability result

$u \in L^{2} (- a, 0, H^{2} (B (σ), ℝ^{N})) \cap H^{1} (- a, 0, L^{2} (B (σ), ℝ^{N}))$ for the solutions $u$ of the class $L^{q} (- T, 0, H^{1, q} (Ω, ℝ^{N})) \cap C^{0, λ} (\bar{Q}, ℝ^{N})$ ( $0 < λ < 1$ , $N$ integer $\geq 1$ ) of the nonlinear parabolic system

$- \sum_{i = 1}^{n} D_{i} a^{i} (X, u, D u) + \frac{\partial u}{\partial t} = B^{0} (X, u, D u)$ with quadratic growth and nonlinearity $q \geq 2$ . This result had been obtained making use of the interpolation theory and an imbedding theorem of Gagliardo-Nirenberg type for functions $u$ belonging to $W^{1, q} \cap C^{0, λ}$ .

Displaying similar documents to “On an over-determined problem of free boundary of a degenerate parabolic equation”

Large time behavior of solutions to a class of doubly nonlinear parabolic equations

A global differentiability result for solutions of nonlinear elliptic problems with controlled growths

Differentiability of weak solutions of nonlinear second order parabolic systems with quadratic growth and nonlinearity <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"> <mrow> <mi>q</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math>

Differentiability of weak solutions of nonlinear second order parabolic systems with quadratic growth and nonlinearity $q \geq 2$