On solutions to the heat equation with the initial condition in the Orlicz—Slobodetskii space

2014 ◽  
Vol 144 (4) ◽  
pp. 787-807 ◽  
Author(s):  
Agnieszka Kałamajska ◽  
Mirosłav Krbec
Keyword(s):  
Sobolev Space ◽  
Boundary Value Problem ◽  
Heat Equation ◽  
Boundary Value ◽  
Orlicz Function ◽  
Initial Condition ◽  
Lipschitz Boundary

We study the boundary-value problem ũt = Δxũ(x,t), ũ(x, 0) = u(x), where x ∈ Ω, t ∈ (0,T), Ω ⊆ ℝn−1 is a bounded Lipschitz boundary domain, u belongs to a certain Orlicz–Slobodetskii space YR,R(Ω). Under certain assumptions on the Orlicz function R, we prove that the solution u belongs to the Orlicz–Sobolev space W1,R(Ω × (0,T)).

Recovery of harmonic functions from partial boundary data respecting internal pointwise values

2017 ◽  
Vol 25 (2) ◽  
Author(s):  
Juliette Leblond ◽  
Dmitry Ponomarev
Keyword(s):  
Boundary Value Problem ◽  
Connected Domain ◽  
Harmonic Functions ◽  
Boundary Data ◽  
Boundary Value ◽  
Lipschitz Boundary ◽  
Simply Connected Domain ◽  
Simply Connected ◽  
Overdetermined Boundary Value Problem

Abstract We consider partially overdetermined boundary-value problem for Laplace PDE in a planar simply connected domain with Lipschitz boundary

scholarly journals Periodic solutions of the boundary value problem for the nonlinear heat equation

1984 ◽  
Vol 30 (1) ◽  
pp. 99-110 ◽  
Author(s):  
M. N. Nkashama ◽  
M. Willem
Keyword(s):  
Boundary Value Problem ◽  
Heat Equation ◽  
Periodic Solutions ◽  
Boundary Value ◽  
Coincidence Degree ◽  
Nonlinear Heat Equation

We prove the existence of generalized periodic solutions of the boundary value problem for the nonlinear heat equation. The proof is based on classical Leray-Schauder's techniques and coincidence degree.

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