scholarly journals On the Parametric Stokes Phenomenon for Solutions of Singularly Perturbed Linear Partial Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-86 ◽  
Author(s):  
Stéphane Malek
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Formal Solution ◽  
Sufficient Conditions ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Stokes Phenomenon ◽  
Partial Differential ◽  
Borel Transform ◽  
Linear Partial Differential Equations

We study a family of singularly perturbed linear partial differential equations with irregular type in the complex domain. In a previous work, Malek (2012), we have given sufficient conditions under which the Borel transform of a formal solution to the above mentioned equation with respect to the perturbation parameter converges near the origin in and can be extended on a finite number of unbounded sectors with small opening and bisecting directions, say , for some integer . The proof rests on the construction of neighboring sectorial holomorphic solutions to the first mentioned equation whose differences have exponentially small bounds in the perturbation parameter (Stokes phenomenon) for which the classical Ramis-Sibuya theorem can be applied. In this paper, we introduce new conditions for the Borel transform to be analytically continued in the larger sectors , where it develops isolated singularities of logarithmic type lying on some half lattice. In the proof, we use a criterion of analytic continuation of the Borel transform described by Fruchard and Schäfke (2011) and is based on a more accurate description of the Stokes phenomenon for the sectorial solutions mentioned above.

scholarly journals On Parametric Gevrey Asymptotics for Singularly Perturbed Partial Differential Equations with Delays

2013 ◽  
Vol 2013 ◽  
pp. 1-18 ◽  
Author(s):  
Alberto Lastra ◽  
Stéphane Malek
Keyword(s):  
Partial Differential Equations ◽  
Power Series ◽  
Differential Equations ◽  
Formal Power Series ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Partial Differential ◽  
Actual Solution ◽  
Gevrey Estimates ◽  
Formal Power

We study a family of singularly perturbed -difference-differential equations in the complex domain. We provide sectorial holomorphic solutions in the perturbation parameter . Moreover, we achieve the existence of a common formal power series in which represents each actual solution and establish -Gevrey estimates involved in this representation. The proof of the main result rests on a new version of the so-called Malgrange-Sibuya theorem regarding -Gevrey asymptotics. A particular Dirichlet like series is studied on the way.

scholarly journals Controllability of systems of linear partial differential equations

2021 ◽  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Schwartz Space ◽  
Complete Controllability ◽  
Controllability Problem ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Necessary And Sufficient ◽  
System 1

In a number of papers, the controllability theory was recently studied. But quite a few of them were devoted to control systems described by ordinary differential equations. In the case of systems described by partial differential equations, they were studied mostly for classical equations of mathematical physics. For example, in papers by G. Sklyar and L. Fardigola, controllability problems were studied for the wave equation on a half-axis. In the present paper, the complete controllability problem is studied for systems of linear partial differential equations with constant coefficients in the Schwartz space of rapidly decreasing functions. Necessary and sufficient conditions for complete controllability are obtained for these systems with distributed control of the special form: u(x,t)=e-αtu(x). To prove these conditions, other necessary and sufficient conditions obtained earlier by the author are applied (see ``Controllability of evolution partial differential equation''. Visnyk of V. N. Karasin Kharkiv National University. Ser. ``Mathematics, Applied Mathematics and Mechanics''. 2016. Vol. 83, p. 47-56). Thus, the system $$\frac{\partial w(x,t)}{\partial t} = P\left(\frac\partial{i\partial x} \right) w(x,t)+ e^{-\alpha t}u(x),\quad t\in[0,T], \ x\in\mathbb R^n, $$ is completely controllable in the Schwartz space if there exists α>0 such that $$\det\left( \int_0^T \exp\big(-t(P(s)+\alpha E)\big)\, dt\right)\neq 0,\quad s\in\mathbb R^N.$$ This condition is equivalent to the following one: there exists $\alpha>0$ such that $$\exp\big(-T(\lambda_j(s)+\alpha)\big)\neq 1 \quad \text{if}\ (\lambda_j(s)+\alpha)\neq0,\qquad s\in\mathbb R^n,\ j=\overline{1,m},$$ where $\lambda_j(s)$, $j=\overline{1,m}$, are eigenvalues of the matrix $P(s)$, $s\in\mathbb R^n$. The particular case of system (1) where $\operatorname{Re} \lambda_j(s)$, $s\in\mathbb R$, $j=\overline{1,m}$, are bounded above or below is studied. These systems are completely controllable. For instance, if the Petrovsky well-posedness condition holds for system (1), then it is completely controllable. Conditions for the existence of a system of the form (1) which is not completely controllable are also obtained. An example of a such kind system is given. However, if a control of the considered form does not exists, then a control of other form solving complete controllability problem may exist. An example illustrating this effect is also given in the paper.

Lyapunov-type inequalities for partial differential equations with 𝑝-Laplacian

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Robert Stegliński
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Nonlinear Partial Differential Equations ◽  
Dirichlet Boundary Conditions ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Lyapunov Inequalities ◽  
Funct Anal

Abstract The aim of this paper is to extend results from [A. Cañada, J. A. Montero and S. Villegas, Lyapunov inequalities for partial differential equations, J. Funct. Anal. 237 (2006), 1, 176–193] about Lyapunov-type inequalities for linear partial differential equations to nonlinear partial differential equations with 𝑝-Laplacian with zero Neumann or Dirichlet boundary conditions.

scholarly journals Slice holomorphic solutions of some directional differential equations with bounded L-index in the same direction

2019 ◽  
Vol 52 (1) ◽  
pp. 482-489 ◽  
Author(s):  
Andriy Bandura ◽  
Oleh Skaskiv ◽  
Liana Smolovyk
Keyword(s):  
Partial Differential Equations ◽  
Continuous Function ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Holomorphic Functions ◽  
Positive Continuous Function ◽  
Partial Differential ◽  
Partial Derivatives ◽  
Fixed Direction

AbstractIn the paper we investigate slice holomorphic functions F : ℂn → ℂ having bounded L-index in a direction, i.e. these functions are entire on every slice {z0 + tb : t ∈ℂ} for an arbitrary z0 ∈ℂn and for the fixed direction b ∈ℂn \ {0}, and (∃m0 ∈ ℤ+) (∀m ∈ ℤ+) (∀z ∈ ℂn) the following inequality holds{{\left| {\partial _{\bf{b}}^mF(z)} \right|} \over {m!{L^m}(z)}} \le \mathop {\max }\limits_{0 \le k \le {m_0}} {{\left| {\partial _{\bf{b}}^kF(z)} \right|} \over {k!{L^k}(z)}},where L : ℂn → ℝ+ is a positive continuous function, {\partial _{\bf{b}}}F(z) = {d \over {dt}}F\left( {z + t{\bf{b}}} \right){|_{t = 0}},\partial _{\bf{b}}^pF = {\partial _{\bf{b}}}\left( {\partial _{\bf{b}}^{p - 1}F} \right)for p ≥ 2. Also, we consider index boundedness in the direction of slice holomorphic solutions of some partial differential equations with partial derivatives in the same direction. There are established sufficient conditions providing the boundedness of L-index in the same direction for every slie holomorphic solutions of these equations.

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