scholarly journals The Existence of Solutions for Local Dirichlet (r(u),s(u))-Problems

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 237
Author(s):  
Calogero Vetro
Keyword(s):  
Polynomial Growth ◽  
Existence Of Solutions ◽  
A Priori Estimates ◽  
A Priori ◽  
Point Theory ◽  
Continuous Functions ◽  
Space Structure ◽  
Laplacian Operator ◽  
Dirichlet Problems ◽  
Priori Estimates

In this paper, we consider local Dirichlet problems driven by the (r(u),s(u))-Laplacian operator in the principal part. We prove the existence of nontrivial weak solutions in the case where the variable exponents r,s are real continuous functions and we have dependence on the solution u. The main contributions of this article are obtained in respect of: (i) Carathéodory nonlinearity satisfying standard regularity and polynomial growth assumptions, where in this case, we use geometrical and compactness conditions to establish the existence of the solution to a regularized problem via variational methods and the critical point theory; and (ii) Sobolev nonlinearity, somehow related to the space structure. In this case, we use a priori estimates and asymptotic analysis of regularized auxiliary problems to establish the existence and uniqueness theorems via a fixed-point argument.

scholarly journals Existence of solutions of a non-variational bi-harmonic system via fixed point theory

Filomat ◽  
2018 ◽  
Vol 32 (12) ◽  
pp. 4113-4130 ◽  
Author(s):  
Idir Mechai ◽  
Metib Alghamdi ◽  
Habib Yazidi
Keyword(s):  
Numerical Solutions ◽  
Existence Of Solutions ◽  
A Priori Estimates ◽  
Fixed Point Theory ◽  
A Priori ◽  
Point Theory ◽  
Estimates Of Solutions ◽  
Priori Estimates ◽  
Theoretical Results ◽  
Harmonic System

We prove existence of a positive solution for a system of non-variational bi-harmonic equations. Furthermore, we give some a priori estimates of solutions and a non-existence result. In addition we compute numerical solutions to illustrate the theoretical results.

scholarly journals Transport equations with nonlocal diffusion and applications to Hamilton–Jacobi equations

2021 ◽  
Author(s):  
Alessandro Goffi
Keyword(s):  
Polynomial Growth ◽  
A Priori Estimates ◽  
A Priori ◽  
Strong Solutions ◽  
Nonlocal Diffusion ◽  
Priori Estimates ◽  
Fokker Planck Equations ◽  
Jacobi Equations ◽  
Hölder Estimates ◽  
Fokker Planck

AbstractWe investigate regularity and a priori estimates for Fokker–Planck and Hamilton–Jacobi equations with unbounded ingredients driven by the fractional Laplacian of order $$s\in (1/2,1)$$ s ∈ ( 1 / 2 , 1 ) . As for Fokker–Planck equations, we establish integrability estimates under a fractional version of the Aronson–Serrin interpolated condition on the velocity field and Bessel regularity when the drift has low Lebesgue integrability with respect to the solution itself. Using these estimates, through the Evans’ nonlinear adjoint method we prove new integral, sup-norm and Hölder estimates for weak and strong solutions to fractional Hamilton–Jacobi equations with unbounded right-hand side and polynomial growth in the gradient. Finally, by means of these latter results, exploiting Calderón–Zygmund-type regularity for linear nonlocal PDEs and fractional Gagliardo–Nirenberg inequalities, we deduce optimal $$L^q$$ L q -regularity for fractional Hamilton–Jacobi equations.

scholarly journals Method of Monotone Solutions for Reaction-Diffusion Equations

2017 ◽  
Vol 63 (3) ◽  
pp. 437-454
Author(s):  
V Volpert ◽  
V Vougalter
Keyword(s):  
Existence Of Solutions ◽  
A Priori Estimates ◽  
A Priori ◽  
Unbounded Domains ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Reaction Diffusion Systems ◽  
Estimates Of Solutions ◽  
Priori Estimates ◽  
Monotone Solutions

Existence of solutions of reaction-diffusion systems of equations in unbounded domains is studied by the Leray-Schauder (LS) method based on the topological degree for elliptic operators in unbounded domains and on a priori estimates of solutions in weighted spaces. We identify some reactiondiffusion systems for which there exist two subclasses of solutions separated in the function space, monotone and non-monotone solutions. A priori estimates and existence of solutions are obtained for monotone solutions allowing to prove their existence by the LS method. Various applications of this method are given.

scholarly journals Existence of positive periodic solutions for super-linear neutral Liénard equation with a singularity of attractive type

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Yu Zhu
Keyword(s):  
Periodic Solutions ◽  
A Priori Estimates ◽  
A Priori ◽  
Continuous Functions ◽  
Liénard Equation ◽  
Positive Periodic Solutions ◽  
Mawhin’S Continuation Theorem ◽  
Priori Estimates ◽  
Lienard Equation ◽  
Singularity Of Attractive Type

Abstract In this paper, the existence of positive periodic solutions is studied for super-linear neutral Liénard equation with a singularity of attractive type $$ \bigl(x(t)-cx(t-\sigma)\bigr)''+f\bigl(x(t) \bigr)x'(t)-\varphi(t)x^{\mu}(t)+ \frac{\alpha(t)}{x^{\gamma}(t)}=e(t), $$ ( x ( t ) − c x ( t − σ ) ) ″ + f ( x ( t ) ) x ′ ( t ) − φ ( t ) x μ ( t ) + α ( t ) x γ ( t ) = e ( t ) , where $f:(0,+\infty)\rightarrow R$ f : ( 0 , + ∞ ) → R , $\varphi(t)>0$ φ ( t ) > 0 and $\alpha(t)>0$ α ( t ) > 0 are continuous functions with T-periodicity in the t variable, c, γ are constants with $|c|<1$ | c | < 1 , $\gamma\geq1$ γ ≥ 1 . Many authors obtained the existence of periodic solutions under the condition $0<\mu\leq1$ 0 < μ ≤ 1 , and we extend the result to $\mu>1$ μ > 1 by using Mawhin’s continuation theorem as well as the techniques of a priori estimates. At last, an example is given to show applications of the theorem.

Well-posedness of a model equation for neurotransmitter diffusion with reactive boundaries

2014 ◽  
Vol 25 (02) ◽  
pp. 195-227 ◽  
Author(s):  
Xavier Raynaud ◽  
Magne Nordaas ◽  
Knut Petter Lehre ◽  
Niels Christian Danbolt
Keyword(s):  
Model Equation ◽  
Existence Of Solutions ◽  
A Priori Estimates ◽  
A Priori ◽  
Well Posedness ◽  
Priori Estimates ◽  
Boundary Cell ◽  
Protein Molecules ◽  
Boundary Reaction ◽  
The Brain

We consider a diffusion equation with reactive boundary conditions. The equation is a model equation for the diffusion of classical neurotransmitters in the tortuous space between cells in the brain. The equation determines the concentration of neurotransmitters such as glutamate and GABA (gamma-aminobutyrate) and the probability for neurotransmitter molecules to be immobilized by binding to protein molecules (receptors and transporters) at the cell boundary (cell membrane). On a regularized problem, we derive a priori estimates. Then, by a compactness argument, we show the existence of solutions. By exploiting the particular structure of the boundary reaction terms, we are able to prove that the solutions are unique and continuous with respect to initial data.

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