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Local Solutions to a Class of Parabolic System Related to the P-Laplacian  [PDF]
Qitong Ou, Huashui Zhan
Advances in Pure Mathematics (APM) , 2016, DOI: 10.4236/apm.2016.612065
Abstract: In this paper, the existence and uniqueness of local solutions to the initial and boundary value problem of a class of parabolic system related to the p-Laplacian are studied. The regularization method is used to construct a sequence of approximation solutions, with the help of monotone iteration technique, then we get the existence of solution of a regularized system. By the use of a standard limiting process, the existence of the local solutions of the system is obtained. Finally, the uniqueness of the solution is also proven.
Existence of Viscosity Solutions to a Parabolic Inhomogeneous Equation Associated with Infinity Laplacian  [PDF]
Fang Liu
Journal of Applied Mathematics and Physics (JAMP) , 2015, DOI: 10.4236/jamp.2015.35061
Abstract: In this paper, we obtain the existence result of viscosity solutions to the initial and boundary value problem for a nonlinear degenerate parabolic inhomogeneous equation of the form \"\", where \"\" denotes infinity Laplacian given by \"\".
Positive and Dead-Core Solutions of Two-Point Singular Boundary Value Problems with -Laplacian  [cached]
Staněk Svatoslav
Advances in Difference Equations , 2010,
Abstract: The paper discusses the existence of positive solutions, dead-core solutions, and pseudo-dead-core solutions of the singular problem , , . Here is a positive parameter, , , , , is singular at and may be singular at .
Quasilinear parabolic problem with $p(x)$-Laplacian: existence, uniqueness of weak solutions and stabilization  [PDF]
Jacques Giacomoni,Sweta Tiwari,Guillaume Warnault
Mathematics , 2015,
Abstract: We discuss the existence and uniqueness of the weak solution of the following quasilinear parabolic equation $u_t-\Delta _{p(x)}u = f(x,u)$ in $ (0,T)\times\Omega$; $u = 0$ on $(0,T)\times\partial\Omega$; $u(0,x)=u_0(x)$ in $\Omega$; involving the $p(x)$-Laplacian operator. Next, we discuss the global behaviour of solutions and in particular some stabilization properties.
Estimates of solutions for the parabolic $p$-Laplacian equation with measure via parabolic nonlinear potentials  [PDF]
Vitali Liskevich,Igor I. Skrypnik,Zeev Sobol
Mathematics , 2012,
Abstract: For weak solutions to the evolutional $p$-Laplace equation with a time-dependent Radon measure on the right hand side we obtain pointwise estimates via a nonlinear parabolic potential.
Positive and Dead-Core Solutions of Two-Point Singular Boundary Value Problems with -Laplacian  [cached]
Svatoslav Staněk
Advances in Difference Equations , 2010, DOI: 10.1155/2010/262854
Abstract: The paper discusses the existence of positive solutions, dead-core solutions, and pseudo-dead-core solutions of the singular problem ( (u′))′=λf(t,u,u′), u(0)-αu′(0)=A, u(T)+βu′(0)+γu′(T)=A. Here λ is a positive parameter, α>0, A>0, β≥0, γ≥0, f is singular at u=0, and f may be singular at u'=0.
Gradient estimates for a degenerate parabolic equation with gradient absorption and applications  [PDF]
Jean-Philippe Bartier,Philippe Lauren?ot
Mathematics , 2007,
Abstract: Qualitative properties of non-negative solutions to a quasilinear degenerate parabolic equation with an absorption term depending solely on the gradient are shown, providing information on the competition between the nonlinear diffusion and the nonlinear absorption. In particular, the limit as time goes to infinity of the mass of integrable solutions is identified, together with the rate of expansion of the support for compactly supported initial data. The persistence of dead cores is also shown. The proof of these results strongly relies on gradient estimates which are first established.
On the viscosity solutions to a degenerate parabolic differential equation  [PDF]
Tilak Bhattacharya,Leonardo Marazzi
Mathematics , 2013, DOI: 10.1007/s10231-014-0427-1
Abstract: In this work, we study some properties of the viscosity solutions to a degenerate parabolic equation involving the non-homogeneous infinity-Laplacian.
C-infinity interfaces of solutions for one-dimensional parabolic p-Laplacian equations
Yoonmi Ham,Youngsang Ko
Electronic Journal of Differential Equations , 1999,
Abstract: We study the regularity of a moving interface $x = zeta (t)$ of the solutions for the initial value problem $$ u_t = left(|u_x|^{p-2}u_x ight)_x quad u(x,0) =u_0 (x),, $$ where $u_0in L^1({Bbb R})$ and $p>2$. We prove that each side of the moving interface is $C^{infty}$.
H?lder gradient estimates for parabolic homogeneous $p$-Laplacian equations  [PDF]
Tianling Jin,Luis Silvestre
Mathematics , 2015,
Abstract: We prove interior H\"older estimates for the spatial gradient of viscosity solutions to the parabolic homogeneous $p$-Laplacian equation \[ u_t=|\nabla u|^{2-p} \mbox{ div} (|\nabla u|^{p-2}\nabla u), \] where $1
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