scholarly journals Existence of solutions for a nonlinear hyperbolic-parabolic equation in a non-cylinder domain

1996 ◽  
Vol 19 (1) ◽  
pp. 151-160 ◽  
Author(s):  
Marcondes Rodrigues Clark
Keyword(s):  
Parabolic Equation ◽  
Galerkin Method ◽  
Weak Solutions ◽  
Symmetric Operator ◽  
Existence Of Solutions ◽  
Global Weak Solutions ◽  
Real Functions ◽  
Compactness Arguments

In this paper, we study the existence of global weak solutions for the equationk2(x)u″+k1(x)u′+A(t)u+|u|ρu=f       (I)in the non-cylinder domainQinRn+1;k1andk2are bounded real functions,A(t)is the symmetric operatorA(t)=−∑i,j=1n∂∂xj(aij(x,t)∂∂xi)           whereaijandfare real functions given inQ. For the proof of existence of global weak solutions we use the Faedo-Galerkin method, compactness arguments and penalization.

scholarly journals Global weak solutions to the generalized mCH equation via characteristics

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Fanqin Zeng ◽  
Yu Gao ◽  
Xiaoping Xue
Keyword(s):  
Weak Solutions ◽  
Radon Measure ◽  
Measure Space ◽  
Blow Up ◽  
Space Time ◽  
Classical Solutions ◽  
Lagrangian Dynamics ◽  
Global Weak Solutions ◽  
Compactness Arguments ◽  
Mollification Method

<p style='text-indent:20px;'>In this paper, we study the generalized modified Camassa-Holm (gmCH) equation via characteristics. We first change the gmCH equation for unknowns <inline-formula><tex-math id="M1">\begin{document}$ (u,m) $\end{document}</tex-math></inline-formula> into its Lagrangian dynamics for characteristics <inline-formula><tex-math id="M2">\begin{document}$ X(\xi,t) $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M3">\begin{document}$ \xi\in\mathbb{R} $\end{document}</tex-math></inline-formula> is the Lagrangian label. When <inline-formula><tex-math id="M4">\begin{document}$ X_\xi(\xi,t)&gt;0 $\end{document}</tex-math></inline-formula>, we use the solutions to the Lagrangian dynamics to recover the classical solutions with <inline-formula><tex-math id="M5">\begin{document}$ m(\cdot,t)\in C_0^k(\mathbb{R}) $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M6">\begin{document}$ k\in\mathbb{N},\; \; k\geq1 $\end{document}</tex-math></inline-formula>) to the gmCH equation. The classical solutions <inline-formula><tex-math id="M7">\begin{document}$ (u,m) $\end{document}</tex-math></inline-formula> to the gmCH equation will blow up if <inline-formula><tex-math id="M8">\begin{document}$ \inf_{\xi\in\mathbb{R}}X_\xi(\cdot,T_{\max}) = 0 $\end{document}</tex-math></inline-formula> for some <inline-formula><tex-math id="M9">\begin{document}$ T_{\max}&gt;0 $\end{document}</tex-math></inline-formula>. After the blow-up time <inline-formula><tex-math id="M10">\begin{document}$ T_{\max} $\end{document}</tex-math></inline-formula>, we use a double mollification method to mollify the Lagrangian dynamics and construct global weak solutions (with <inline-formula><tex-math id="M11">\begin{document}$ m $\end{document}</tex-math></inline-formula> in space-time Radon measure space) to the gmCH equation by some space-time BV compactness arguments.</p>

Global weak solutions for elastic equations with damping and different end states

1998 ◽  
Vol 128 (4) ◽  
pp. 797-807 ◽  
Author(s):  
Tao Luo ◽  
Tong Yang
Keyword(s):  
Parabolic Equation ◽  
Weak Solutions ◽  
Compensated Compactness ◽  
Global Weak Solutions ◽  
Compactness Method

In this paper, we study global weak solutions for elastic equations with damping using the compensated compactness method. When the two end states at ± ∞ are not equal, the selfsimilar solutions for the corresponding parabolic equation are used to get the entropic estimates for both the L∞ and L2 cases.

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 Equivalence of viscosity and weak solutions for a p-parabolic equation

2021 ◽  
Author(s):  
Jarkko Siltakoski
Keyword(s):  
Parabolic Equation ◽  
Weak Solutions ◽  
Lower Semicontinuity ◽  
Lower Semicontinuous ◽  
Relationship Of ◽  
The Relationship

AbstractWe study the relationship of viscosity and weak solutions to the equation $$\begin{aligned} \smash {\partial _{t}u-\varDelta _{p}u=f(Du)}, \end{aligned}$$ ∂ t u - Δ p u = f ( D u ) , where $$p>1$$ p > 1 and $$f\in C({\mathbb {R}}^{N})$$ f ∈ C ( R N ) satisfies suitable assumptions. Our main result is that bounded viscosity supersolutions coincide with bounded lower semicontinuous weak supersolutions. Moreover, we prove the lower semicontinuity of weak supersolutions when $$p\ge 2$$ p ≥ 2 .

scholarly journals $L^{\infty }$ decay estimates of solutions of nonlinear parabolic equation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Hui Wang ◽  
Caisheng Chen
Keyword(s):  
Parabolic Equation ◽  
Weak Solutions ◽  
Nonlinear Parabolic Equation ◽  
Divergence Form ◽  
Decay Estimates ◽  
Laplacian Equation ◽  
Estimates Of Solutions ◽  
Nonlinear Parabolic ◽  
Doubly Nonlinear ◽  
Doubly Nonlinear Parabolic Equation

AbstractIn this paper, we are interested in $L^{\infty }$ L ∞ decay estimates of weak solutions for the doubly nonlinear parabolic equation and the degenerate evolution m-Laplacian equation not in the divergence form. By a modified Moser’s technique we obtain $L^{\infty }$ L ∞ decay estimates of weak solutiona.

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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