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This paper is devoted to the study of the following fourth order parabolic equation ∂tu + ∂x4u = f in the non-necessarily rectangular domain

In this paper, we investigate a class of initial boundary value problems for a class of the Kawahara equation. We prove existence of at least one solution and existence of at least one nonnegative solution. The arguments are based upon of recent theoretical results. [ABSTRACT FROM AUTHOR]

We will prove well posedness and regularity results for the bi-dimensional heat equation, subject to mixed Dirichlet-Neumann type boundary conditions on the parabolic boundary of an unbounded (in one space variable direction) time-dependent domain. Our results are proved in anisotropic Hilbertian Sobolev spaces by using the domain decomposition method. This work complements the results obtained in [13] in the one-space variable case. [ABSTRACT FROM AUTHOR]

This paper is devoted to the analysis of the following linear 2m-th order parabolic equation ∂tu+(-1) m ∑ N k=1 ∂ 2m xk u = f, subject to Dirichlet type condition ∂ l νu = 0, l = 0,1,...,m-1, on the lateral boundary, where m is a positive integer. The right-hand side f of the equation is taken in the Lebesgue space L p, 1 < p < +∞. The problem is set in a domain of the form Q = (t, x1,..., xN) ∈ R N+1 : 0 ≤ q x 2 1 +...+x 2 N < t α with α > 1/2m. [ABSTRACT FROM AUTHOR]

This paper is devoted to the analysis of the boundary value problem ∂ t ⁡ u - Δ ⁢ u = f {\partial_{t}u-\Delta u=f} , with an N-dimensional space variable, subject to a Dirichlet–Robin type boundary condition on the lateral boundary of the domain. The problem is settled in a noncylindrical domain of the form Q = { (t , x 1) ∈ ℝ 2 : 0 < t < T , φ 1 (t) < x 1 < φ 2 (t) } × ∏ i = 1 N - 1 ] 0 , b i [ Q=\{(t,x_{1})\in\mathbb{R}^{2}:04]. [ABSTRACT FROM AUTHOR]

This paper is devoted to the analysis of the boundary value problem ∂ t ⁡ u - Δ ⁢ u = f {\partial_{t}u-\Delta u=f} , with an N-dimensional space variable, subject to a Dirichlet–Robin type boundary condition on the lateral boundary of the domain. The problem is settled in a noncylindrical domain of the form Q = { (t , x 1) ∈ ℝ 2 : 0 < t < T , φ 1 (t) < x 1 < φ 2 (t) } × ∏ i = 1 N - 1 ] 0 , b i [ Q=\{(t,x_{1})\in\mathbb{R}^{2}:04]. [ABSTRACT FROM AUTHOR]

In this paper, we use the domain decomposition method to prove well-posedness and smoothness results in anisotropic weighted Sobolev spaces for a multidimensional high-order parabolic equation set in conical time-dependent domains of [ABSTRACT FROM AUTHOR]

In this paper we give new results of existence, uniqueness and maximal regularity of the solution to the N-dimensional heat equation ∂tu ― Δu = f, with Cauchy-Dirichlet boundary conditions in a time-dependent domain. [ABSTRACT FROM AUTHOR]

This paper deals with the existence, uniqueness and maximal regularity of a solution to the multidimensional parabolic equation of order 2 m (with ) defined on a type of the prism of ℝN+1, which is a nonstandard domain since its base shrinks at t = 0 (). The equation is associated with Cauchy-Dirichlet boundary conditions and the right-hand side term f is taken in a weighted Lebesgue space. More precisely, we look for minimal conditions on the functions and under which the solution is regular. For this purpose, the domain decomposition method is employed. This work is an extension of the one space variable case studied in and of the case m = 1 studied in . [ABSTRACT FROM AUTHOR]

This paper deals with the existence, uniqueness and maximal regularity of a solution to the multidimensional parabolic equation of order 2 m (with ) defined on a type of the prism of ℝN+1, which is a nonstandard domain since its base shrinks at t = 0 (). The equation is associated with Cauchy-Dirichlet boundary conditions and the right-hand side term f is taken in a weighted Lebesgue space. More precisely, we look for minimal conditions on the functions and under which the solution is regular. For this purpose, the domain decomposition method is employed. This work is an extension of the one space variable case studied in and of the case m = 1 studied in . [ABSTRACT FROM AUTHOR]

In this paper, we look for sufficient conditions on the lateral surface of the domain and on the coefficients of the boundary conditions of a N-space dimensional linear parabolic equation, in order to obtain existence, uniqueness and maximal regularity of the solution in a Hilbertian anisotropic Sobolev space when the right hand side of the equation is in a Lebesgue space. This work is an extension of solvability results obtained for a second order parabolic equation, set in a non-regular domain of ℝ³ obtained in [1], to the case where the domain is cylindrical, not with respect to the time variable, but with respect to N space variables, N > 1. [ABSTRACT FROM AUTHOR]

In this work, we give new results of existence, uniqueness and maximal regularity of a solution to the following two space linear fourth order parabolic equation $${\partial_{t}u + \partial _{x_{1}}^{4}u + \partial _{x_{2}}^{4}u = f}$$ , set in a domain $${Q = \left\{\left( t, x_{1} \right) \in \mathbb{R}^{2} : 0 < t < T; \varphi _{1}\left( t \right) < x_{1} < \varphi _{2}\left(t \right) \right\}\times \left] 0, b \right[ }$$ of $${\mathbb{R} ^{3}}$$ , with Cauchy-Dirichlet boundary conditions, under some assumptions on the functions ( $${\varphi _{i}}$$ ). The right-hand side f of the equation is taken in a weighted Lebesgue space. One of the main issues of this work is that the domain Q can possibly be nonregular, for instance, the singular case where $${\varphi_{1} \left( 0 \right) = \varphi _{2}\left( 0 \right) }$$ is allowed. The method used is based on the approximation of the domain Q by a sequence of subdomains $${\left( Q_{n} \right) _{n}}$$ which can be transformed into regular domains. This work is an extension of the one space variable case studied in []. [ABSTRACT FROM AUTHOR]

This paper is concerned with the extension of solvability results obtained for a fourth order parabolic equation, set in a nonregular domain of ℝ³ obtained in [1], to the case where the domain is cylindrical, not with respect to the time variable, but with respect to N space variables, N > 1. More precisely, we determine optimal conditions on the shape of the boundary of a (N + 1)-dimensional domain, N > 1, under which the solution is regular. [ABSTRACT FROM AUTHOR]

This paper is devoted to the analysis of the N-space dimensional heat equation, subject to Cauchy-Dirichlet boundary conditions. The problem is set in a symmetric conical type domain. More precisely, we look for sufficient conditions on the lateral boundary of the domain, as weak as possible in order to obtain the maximal regularity of the solution in an anisotropic Hilbertian Sobolev space. For this purpose, the domain decomposition method is employed. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

New results on the existence, uniqueness and maximal regularity of a solution are given for a parabolic equation set in a non-regular domain of ℝN+1, with Cauchy-Dirichlet boundary conditions, under some assumptions on the functions and . The right-hand side term of the equation is taken in . The method used is based on the approximation of the domain Q by a sequence of sub-domains which can be transformed into regular domains. This work is an extension of the two space variables case studied in [Differ. Equ. Appl. 2 (2010), no. 2, 251-263]. [ABSTRACT FROM AUTHOR]

New results on the existence, uniqueness and maximal regularity of a solution are given for a parabolic equation set in a non-regular domain of ℝN+1, with Cauchy-Dirichlet boundary conditions, under some assumptions on the functions and . The right-hand side term of the equation is taken in . The method used is based on the approximation of the domain Q by a sequence of sub-domains which can be transformed into regular domains. This work is an extension of the two space variables case studied in [Differ. Equ. Appl. 2 (2010), no. 2, 251-263]. [ABSTRACT FROM AUTHOR]

This article is devoted to the analysis of a two-space dimensional linear parabolic equation, subject to Cauchy-Dirichlet boundary conditions. The problem is set in a conical type domain and the right hand side term of the equation is taken in a Lebesgue space. One of the main issues of this work is that the domain can possibly be non regular. This work is an extension of the symmetric case studied in Sadallah [13]. [ABSTRACT FROM AUTHOR]

In this article, we study the parabolic equation ∂tu-c²(t)∂x²u = f in the non-necessarily rectangular domain Ω = {(t, x) ∈ ℝ² : 0 < t < T, φ1(t) < x < ψ2(t)}. The boundary conditions are of Robin type, while the right-hand side lies in the Lebesgue space L²(Ω). Our aim is to find conditions on c and the functions (ψi)i=1,2 such that the solution belongs to the anisotropic Sobolev space H1,2(Ω) = {u ∈ L²(Ω) : ∂tu, ∂xu, ∂x²u ∈ L²(Ω)}. For goal we use the method of approximation of domains. [ABSTRACT FROM AUTHOR]

In this paper, we use the domain decomposition method to prove well-posedness and smoothness results in anisotropic weighted Sobolev spaces for a multidimensional high-order parabolic equation set in conical time-dependent domains of [ABSTRACT FROM AUTHOR]