An example of the equation $u_t = u_{xx} + f(x, t, u)$ with distinct maximum and minimum solutions of a mixed problem

W. Mlak

Displaying similar documents to “An example of the equation $u_{t} = u_{x x} + f (x, t, u)$ with distinct maximum and minimum solutions of a mixed problem”

On the existence of exactly one solution of integral equations in the space $L^{P}$ with a mixed norm

Bogdan Rzepecki (1975)

Annales Polonici Mathematici

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Minimality properties of Tsirelson type spaces

Denka Kutzarova, Denny H. Leung, Antonis Manoussakis, Wee-Kee Tang (2008)

Studia Mathematica

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We study minimality properties of partly modified mixed Tsirelson spaces. A Banach space with a normalized basis $(e_{k})$ is said to be subsequentially minimal if for every normalized block basis $(x_{k})$ of $(e_{k})$ , there is a further block basis $(y_{k})$ of $(x_{k})$ such that $(y_{k})$ is equivalent to a subsequence of $(e_{k})$ . Sufficient conditions are given for a partly modified mixed Tsirelson space to be subsequentially minimal, and connections with Bourgain’s ℓ¹-index are established. It is also shown that a large class of...

Subspaces $H_{E}^{p, q, α}$ of mixed-norm spaces $H^{p, q, α}$ of holomorphic functions

M. Jevtić (1985)

Matematički Vesnik

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Convexities of Gaussian integral means and weighted integral means for analytic functions

Haiying Li, Taotao Liu (2019)

Czechoslovak Mathematical Journal

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We first show that the Gaussian integral means of $f : ℂ \to ℂ$ (with respect to the area measure $e^{- {α | z |}^{2}} d A (z)$ ) is a convex function of $r$ on $(0, \infty)$ when $α \leq 0$ . We then prove that the weighted integral means $A_{α, β} (f, r)$ and $L_{α, β} (f, r)$ of the mixed area and the mixed length of $f (r 𝔻)$ and $\partial f (r 𝔻)$ , respectively, also have the property of convexity in the case of $α \leq 0$ . Finally, we show with examples that the range $α \leq 0$ is the best possible.