scholarly journals Well-posed results for nonlocal fractional parabolic equation involving Caputo-Fabrizio operator

2022 ◽  
Vol 26 (04) ◽  
pp. 357-367
Author(s):  
T. T. Phong ◽  
L. D. Long
Keyword(s):  
Parabolic Equation ◽  
Well Posed

Blow-up of solutions of a nonlinear parabolic equation in damage mechanics

1997 ◽  
Vol 8 (1) ◽  
pp. 89-123 ◽  
Author(s):  
MICHIEL BERTSCH ◽  
PAOLO BISEGNA
Keyword(s):  
Parabolic Equation ◽  
Damage Mechanics ◽  
Blow Up ◽  
Quasilinear Equation ◽  
Nonlinear Parabolic ◽  
Nonzero Measure ◽  
Degenerate Parabolic ◽  
Theory Of Damage ◽  
Well Posed ◽  
Nonlinear Degenerate

A fully nonlinear, degenerate parabolic equation arising in the theory of damage mechanics is shown to be well-posed. Its solutions blow up in finite time and, under suitable conditions on the initial configuration, the blow-up set, corresponding to the portion of the material which breaks at the blow-up time, is an interval of nonzero measure. In a special but physically relevant case the problem reduces to the study of the blow-up set of solutions of the quasilinear equation

A NONLINEAR BACKWARD PARABOLIC PROBLEM: REGULARIZATION BY QUASI-REVERSIBILITY AND ERROR ESTIMATES

2011 ◽  
Vol 04 (01) ◽  
pp. 145-161 ◽  
Author(s):  
Nguyen Huy Tuan ◽  
Pham Hoang Quan ◽  
Dang Duc Trong ◽  
Nguyen Do Minh Nhat
Keyword(s):  
Exact Solution ◽  
Parabolic Equation ◽  
Small Parameter ◽  
Parabolic Problem ◽  
Unbounded Operator ◽  
Numerical Tests ◽  
Nonlinear Parabolic ◽  
Time Problem ◽  
Ill Posed ◽  
Well Posed

In this paper, we consider an inverse time problem for a nonlinear parabolic equation in the form ut + Au(t) = f(t, u(t)), u(T) = φ, where A is a positive self-adjoint unbounded operator and f is a Lipschitz function. As known, it is ill-posed. Using a quasi-reversibility method, we shall construct regularization solutions depended on a small parameter ϵ. We show that the regularized problem is well-posed and that their solution uϵ(t) converges on [0, T] to the exact solution u(t). This paper extends the work by Dinh Nho Hao et al. [8] to nonlinear ill-posed problems. Some numerical tests illustrate that the proposed method is feasible and effective.

Hypothesis on the Solvability of Parabolic Equations with nonlocal Condition

2002 ◽  
Vol 7 (1) ◽  
pp. 93-104 ◽  
Author(s):  
Mifodijus Sapagovas
Keyword(s):  
Parabolic Equation ◽  
Parabolic Equations ◽  
Existence And Uniqueness ◽  
Nonlocal Condition ◽  
Nonlocal Conditions ◽  
Numerical Algorithms ◽  
Uniqueness Of The Solution

Numerous and different nonlocal conditions for the solvability of parabolic equations were researched in many articles and reports. The article presented analyzes such conditions imposed, and observes that the existence and uniqueness of the solution of parabolic equation is related mainly to ”smallness” of functions, involved in nonlocal conditions. As a consequence the hypothesis has been made, stating the assumptions on functions in nonlocal conditions are related to numerical algorithms of solving parabolic equations, and not to the parabolic equation itself.

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