Local Existence and Uniqueness of Sobolev Type Semilinear Initial Value Problems in Banach Spaces

2006 ◽  
Vol 9 (2) ◽  
pp. 143-148
Author(s):  
Akram M. Al-Abood ◽  
◽  
Manaf A. Salah ◽  
Keyword(s):  
Banach Spaces ◽  
Existence And Uniqueness ◽  
Initial Value Problems ◽  
Local Existence ◽  
Sobolev Type ◽  
Initial Value ◽  
Local Existence And Uniqueness

scholarly journals Local existence in time of small solutions to the elliptic-hyperbolic Davey-Stewartson system in the usual Sobolev space

1997 ◽  
Vol 40 (3) ◽  
pp. 563-581 ◽  
Author(s):  
Nakao Hayashi ◽  
Hitoshi Hirata
Keyword(s):  
Sobolev Space ◽  
Initial Value Problem ◽  
Existence And Uniqueness ◽  
Local Existence ◽  
Initial Value ◽  
Hyperbolic Case ◽  
Local Existence And Uniqueness ◽  
Usual Sobolev Space

We study the initial value problem to the Davey-Stewartson system for the elliptic-hyperbolic case in the usual Sobolev space. We prove local existence and uniqueness H5/2 with a condition such that the L2 norm of the data is sufficiently small.

scholarly journals Exponential decay of solutions with \(L^{p}\)-norm for a class to semilinear wave equation with damping and source terms

2020 ◽  
Vol 4 (2) ◽  
pp. 123-131
Author(s):  
Amar Ouaoua ◽  
◽  
Messaoud Maouni ◽  
Aya Khaldi ◽  
◽  
...  
Keyword(s):  
Wave Equation ◽  
Lyapunov Function ◽  
Existence And Uniqueness ◽  
Local Existence ◽  
Local Solution ◽  
Initial Value ◽  
Local Existence And Uniqueness ◽  
Decay Of Solutions ◽  
Uniqueness Of The Solution ◽  
Damping And Source Terms

In this paper, we consider an initial value problem related to a class of hyperbolic equation in a bounded domain is studied. We prove local existence and uniqueness of the solution by using the Faedo-Galerkin method and that the local solution is global in time. We also prove that the solutions with some conditions exponentially decay. The key tool in the proof is an idea of Haraux and Zuazua with is based on the construction of a suitable Lyapunov function.

scholarly journals Existence results for second order functional differential and integrodifferential inclusions in Banach spaces

2001 ◽  
Vol 32 (4) ◽  
pp. 315-325
Author(s):  
M. Benchohra ◽  
S. K. Ntouyas
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Banach Spaces ◽  
Initial Value Problems ◽  
Existence Of Solutions ◽  
Existence Results ◽  
Second Order ◽  
Initial Value ◽  
Functional Differential ◽  
Condensing Maps

In this paper we investigate the existence of solutions on a compact interval to second order initial value problems for functional differential and integrodifferential inclusions in Banach spaces. We shall make use of a fixed point theorem for condensing maps due to Martelli.

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