An Elliptic Equation with Blowing-Up Diffusion and Data in L1: Existence and Uniqueness

2003 ◽  
Vol 13 (09) ◽  
pp. 1351-1377 ◽  
Author(s):  
Concepción García Vázquez ◽  
Francisco Ortegón Gallego
Keyword(s):  
Elliptic Equation ◽  
Existence And Uniqueness ◽  
Positive Function ◽  
Renormalized Solutions ◽  
Diffusion Matrix ◽  
Existence And Uniqueness Results ◽  
Nonlinear Elliptic ◽  
Uniqueness Results ◽  
Blowing Up ◽  
The Right

We establish some existence and uniqueness results for a nonlinear elliptic equation. The problem has a diffusion matrix A(x, u) such that A(x, s)ξξ ≥ β(s)|ξ|2, with β : (s0, + ∞) ↦ ℝ a continuous, strictly positive function which goes to infinity when s is near s0. On the other hand, [Formula: see text]. Also, the right-hand side f belongs to L1(Ω). We make use of the concept of renormalized solutions adapted to our problem.

scholarly journals A source inverse problem for the pseudo–parabolic equation with the fractional Sturm–Liouville operator

2020 ◽  
Vol 100 (4) ◽  
pp. 143-151
Author(s):  
Daurenbek Serikbaev ◽  
◽  
Niyaz Tokmagambetov ◽  
◽  
◽  
...  
Keyword(s):  
Parabolic Equation ◽  
Inverse Problems ◽  
Existence And Uniqueness ◽  
Liouville Operator ◽  
Mineral Exploration ◽  
Fourier Method ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Sturm Liouville ◽  
The Right

A class of inverse problems for restoring the right-hand side of the pseudo-parabolic equation with one fractional Sturm–Liouville operator is considered. In this paper, we prove the existence and uniqueness results of the solutions using by the variable separation method that is to say the Fourier method. We are especially interested in proving the existence and uniqueness of the solutions in the abstract setting of Hilbert spaces. The mentioned results are presented as well as for the Caputo time fractional pseudoparabolic equation. There are many cases in which practical needs lead to problems determining the coefficients or the right side of a differential equation from some available decision data. These are called inverse problems of mathematical physics. Inverse problems arise in various areas of human activity, such as seismology, mineral exploration, biology, medicine, industrial quality control goods, and so on. All these circumstances put the inverse problems among the important problems of modern mathematics.

Problèmes elliptiques du 2ème ordre non sous forme divergence

1979 ◽  
Vol 84 (3-4) ◽  
pp. 263-271 ◽  
Author(s):  
Pierre-Louis Lions
Keyword(s):  
Existence And Uniqueness ◽  
Elliptic Boundary ◽  
Linear Part ◽  
Divergence Form ◽  
Elliptic Boundary Value ◽  
Existence And Uniqueness Results ◽  
Nonlinear Elliptic ◽  
Uniqueness Results ◽  
Order Elliptic Operator ◽  
Nonlinear Elliptic Boundary

SynopsisWe prove existence and uniqueness results for the solution of nonlinear elliptic boundary value problems, where the linear part of the equation is given by a second-order elliptic operator not in divergence form.

scholarly journals Multi-term fractional differential equations with nonlocal boundary conditions

2018 ◽  
Vol 16 (1) ◽  
pp. 1519-1536
Author(s):  
Bashir Ahmad ◽  
Najla Alghamdi ◽  
Ahmed Alsaedi ◽  
Sotiris K. Ntouyas
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Fractional Differential ◽  
The Given

AbstractWe introduce and study a new kind of nonlocal boundary value problems of multi-term fractional differential equations. The existence and uniqueness results for the given problem are obtained by applying standard fixed point theorems. We also construct some examples for demonstrating the application of the main results.

scholarly journals BSDEs with polynomial growth generators

2000 ◽  
Vol 13 (3) ◽  
pp. 207-238 ◽  
Author(s):  
Philippe Briand ◽  
René Carmona
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Existence And Uniqueness ◽  
Polynomial Growth ◽  
Probabilistic Representation ◽  
Existence And Uniqueness Results ◽  
State Variable ◽  
Cahn Equation ◽  
Uniqueness Results ◽  
Terminal Time

In this paper, we give existence and uniqueness results for backward stochastic differential equations when the generator has a polynomial growth in the state variable. We deal with the case of a fixed terminal time, as well as the case of random terminal time. The need for this type of extension of the classical existence and uniqueness results comes from the desire to provide a probabilistic representation of the solutions of semilinear partial differential equations in the spirit of a nonlinear Feynman-Kac formula. Indeed, in many applications of interest, the nonlinearity is polynomial, e.g, the Allen-Cahn equation or the standard nonlinear heat and Schrödinger equations.

Nontrivial solutions of non-autonomous dirichlet fractional discrete problems

2020 ◽  
Vol 23 (4) ◽  
pp. 980-995
Author(s):  
Alberto Cabada ◽  
Nikolay Dimitrov
Keyword(s):  
Existence And Uniqueness ◽  
Point Boundary ◽  
Nontrivial Solutions ◽  
Fractional Difference ◽  
Nonlinear Part ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Fractional Difference Equation ◽  
Discrete Problems ◽  
Series Of Functions

AbstractIn this paper, we introduce a two-point boundary value problem for a finite fractional difference equation with a perturbation term. By applying spectral theory, an associated Green’s function is constructed as a series of functions and some of its properties are obtained. Under suitable conditions on the nonlinear part of the equation, some existence and uniqueness results are deduced.

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