scholarly journals Existence of weak solutions for viscoelastic hyperbolic equations with variable exponents

2013 ◽  
Vol 2013 (1) ◽  
Author(s):  
Yunzhu Gao ◽  
Wenjie Gao
Keyword(s):  
Weak Solutions ◽  
Hyperbolic Equations ◽  
Variable Exponents ◽  
Existence Of Weak Solutions

scholarly journals Existence of weak solutions to a certain homogeneous parabolic Neumann problem involving variable exponents and cross-diffusion

2020 ◽  
Vol 6 (2) ◽  
pp. 685-709
Author(s):  
Gurusamy Arumugam ◽  
André H. Erhardt
Keyword(s):  
Neumann Problem ◽  
Weak Solutions ◽  
Type Inequality ◽  
Neumann Boundary ◽  
Stability Estimate ◽  
Energy Estimates ◽  
Variable Exponents ◽  
Existence Of Weak Solutions ◽  
Cross Diffusion ◽  
Time Variables

Abstract This paper deals with a homogeneous Neumann problem of a nonlinear diffusion system involving variable exponents dependent on spatial and time variables and cross-diffusion terms. We prove the existence of weak solutions using Galerkin’s approximation and we derive suitable energy estimates. To this end, we establish the needed Poincaré type inequality for variable exponents related to the Neumann boundary problem. Furthermore, we show that the investigated problem possesses a unique weak solution and satisfies a stability estimate, provided some additional assumptions are fulfilled. In addition, we show under which conditions the solution is nonnegative.

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.

scholarly journals On a degenerate parabolic equation with Newtonian fluid∼non-Newtonian fluid mixed type

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Sujun Weng
Keyword(s):  
Parabolic Equation ◽  
Newtonian Fluid ◽  
Weak Solutions ◽  
Mixed Type ◽  
Degenerate Parabolic Equation ◽  
Type Equation ◽  
Existence Of Weak Solutions ◽  
Degenerate Parabolic ◽  
The Stability ◽  
Increasing Function

AbstractWe study the existence of weak solutions to a Newtonian fluid∼non-Newtonian fluid mixed-type equation $$ {u_{t}}= \operatorname{div} \bigl(b(x,t){ \bigl\vert {\nabla A(u)} \bigr\vert ^{p(x) - 2}}\nabla A(u)+\alpha (x,t)\nabla A(u) \bigr)+f(u,x,t). $$ u t = div ( b ( x , t ) | ∇ A ( u ) | p ( x ) − 2 ∇ A ( u ) + α ( x , t ) ∇ A ( u ) ) + f ( u , x , t ) . We assume that $A'(s)=a(s)\geq 0$ A ′ ( s ) = a ( s ) ≥ 0 , $A(s)$ A ( s ) is a strictly increasing function, $A(0)=0$ A ( 0 ) = 0 , $b(x,t)\geq 0$ b ( x , t ) ≥ 0 , and $\alpha (x,t)\geq 0$ α ( x , t ) ≥ 0 . If $$ b(x,t)=\alpha (x,t)=0,\quad (x,t)\in \partial \Omega \times [0,T], $$ b ( x , t ) = α ( x , t ) = 0 , ( x , t ) ∈ ∂ Ω × [ 0 , T ] , then we prove the stability of weak solutions without the boundary value condition.

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