scholarly journals The Local Strong and Weak Solutions for a Nonlinear Dissipative Camassa-Holm Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-15
Author(s):  
Shaoyong Lai
Keyword(s):  
Sobolev Space ◽  
Differential Equations ◽  
Weak Solutions ◽  
Lower Order ◽  
Well Posedness ◽  
Sufficient Condition ◽  
Existence Of Weak Solutions ◽  
Abstract Differential Equations ◽  
Holm Equation

Using the Kato theorem for abstract differential equations, the local well-posedness of the solution for a nonlinear dissipative Camassa-Holm equation is established in spaceC([0,T),Hs(R))∩C1([0,T),Hs-1(R))withs>3/2. In addition, a sufficient condition for the existence of weak solutions of the equation in lower order Sobolev spaceHs(R)with1≤s≤3/2is developed.

scholarly journals The Local Strong and Weak Solutions for a Generalized Novikov Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Meng Wu ◽  
Yue Zhong
Keyword(s):  
Sobolev Space ◽  
Differential Equations ◽  
Strong Solution ◽  
Weak Solutions ◽  
Lower Order ◽  
Well Posedness ◽  
Existence Of Weak Solutions ◽  
Abstract Differential Equations ◽  
Novikov Equation ◽  
A Generalized Novikov Equation

The Kato theorem for abstract differential equations is applied to establish the local well-posedness of the strong solution for a nonlinear generalized Novikov equation in spaceC([0,T),Hs(R))∩C1([0,T),Hs-1(R))withs>(3/2). The existence of weak solutions for the equation in lower-order Sobolev spaceHs(R)with1≤s≤(3/2)is acquired.

scholarly journals Existence of weak solutions to SPDEs with fractional Laplacian and non-Lipschitz coefficients

2021 ◽  
Author(s):  
Shohei Nakajima
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Weak Solutions ◽  
Stochastic Partial Differential Equations ◽  
Fractional Laplacian ◽  
Existence Of Solutions ◽  
Partial Differential ◽  
Continuity Theorem ◽  
Existence Of Weak Solutions ◽  
One Dimensional

AbstractWe prove existence of solutions and its properties for a one-dimensional stochastic partial differential equations with fractional Laplacian and non-Lipschitz coefficients. The method of proof is eatablished by Kolmogorov’s continuity theorem and tightness arguments.

Weak Solutions for Fractional Differential Equations via Henstock–Kurzweil–Pettis Integrals

2020 ◽  
Vol 21 (2) ◽  
pp. 135-145 ◽  
Author(s):  
Haide Gou ◽  
Yongxiang Li
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Fractional Differential Equations ◽  
Weak Solutions ◽  
Measure Of Noncompactness ◽  
Pettis Integral ◽  
Existence Of Weak Solutions ◽  
Fractional Differential

AbstractIn this paper, we used Henstock–Kurzweil–Pettis integral instead of classical integrals. Using fixed point theorem and weak measure of noncompactness, we study the existence of weak solutions of boundary value problem for fractional integro-differential equations in Banach spaces. Our results generalize some known results. Finally, an example is given to demonstrate the feasibility of our conclusions.

MOUNTAIN PASS TYPE SOLUTIONS FOR QUASILINEAR ELLIPTIC INCLUSIONS

2002 ◽  
Vol 04 (04) ◽  
pp. 607-637 ◽  
Author(s):  
PH. CLÉMENT ◽  
M. GARCÍA-HUIDOBRO ◽  
R. MANÁSEVICH
Keyword(s):  
Sobolev Space ◽  
Bounded Domain ◽  
Weak Solutions ◽  
Mountain Pass Theorem ◽  
Maximal Monotone ◽  
Mountain Pass ◽  
Inclusion Problem ◽  
Existence Of Weak Solutions ◽  
Smooth Case ◽  
Quasilinear Elliptic

We establish the existence of weak solutions to the inclusion problem [Formula: see text] where Ω is a bounded domain in ℝN, [Formula: see text], and ψ ∊ ℝ × ℝ is a maximal monotone odd graph. Under suitable conditions on ψ, g (which reduce to subcritical and superlinear conditions in the case of powers) we obtain the existence of non-trivial solutions which are of mountain pass type in an appropriate not necessarily reflexive Orlicz Sobolev space. The proof is based on a version of the Mountain Pass Theorem for a non-smooth case.

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