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PARABOLIC EQUATIONS WITH NONLINEAR SINGULARITIES

, 2009
Martínez-Aparicio, Pedro J., Petitta, Francesco   +1 more
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Some Degenerate Nonlinear Parabolic Equations

Some Degenerate Nonlinear Parabolic Equations
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A Nonlinear Parabolic Equation with Noise

Potential Analysis, 2000
The authors consider the semilinear stochastic parabolic partial differential equation (PDE, in short) with multiplicative white noise \[ \partial_t \varphi+ \sum^d_{k=1} \partial_k\bigl(f(t,x, \varphi(t,x) \bigr)= \nu\sum^d_{k=1} \partial^2_{x_k} \varphi+\sigma (t)\varphi(t,x)\dot W_t, \quad (t,x) \in[0,T] \times\mathbb{R}^n, \] where the stochastic ...
Benth, Fred Espen, Gjessing, Håkon K.
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Nonlinear Degenerate Parabolic Equations

Acta Mathematica Hungarica, 1997
The author proves the existence of weak solutions of the nonlinear degenerate parabolic initial-boundary value problem \[ {{\partial u}\over{\partial t}} - \sum_{i=1}^N D_iA_i(x,t,u,Du) + A_0(x,t,u,Du) = f(x,t)\quad\text{ in }\Omega\times(0,T), \] \[ u(x,0) = u_0(x)\quad \hbox{ in }\Omega, \] in the space \(L^p(0,T,W^{1,p}_0(v,\Omega))\), where ...
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Nonlinear Parabolic Equations

1985
In this chapter we wish to consider the nonlinear parabolic equation $$ \begin{gathered} {{u}_{t}} - {{u}_{{xx}}} = g(u) \hfill \\ u(0, t) = u(1,t) = 0 \hfill \\ \mathop{{\lim }}\limits_{{t \to 0}} u(x,t) = f(x) \hfill \\ \end{gathered} $$ (1) We wish to establish an analogue of the classical PoincareLyapunov theorem:If \(g\left( u \right ...
Richard Bellman, George Adomian
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STABILIZATION OF NONLINEAR PARABOLIC EQUATIONS

IFAC Proceedings Volumes, 1983
Abstract We are concerned with the possibility of constructing implementable feedback control laws to stabilize ů + Au = f(u), primarily through the boundary conditions. Semigroup methods are employed to reduce the semi- linear problem to a linear one, to show stabilizability of certain parabolic problems by feedback and, finally, to show for the one-
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Nonlinear Degenerate Parabolic Equation with Nonlinear Boundary Condition

Acta Mathematica Sinica, English Series, 2005
The authors study the existence and nonexistence of global positive solutions to the following nonlinear parabolic equation with nonlinear boundary conditions \[ \begin{aligned} & (u^k)_t = \Delta_mu,\quad x \in\Omega, \quad t > 0,\\ & \nabla_m u\cdot\nu = u^{\alpha},\quad x\in \partial\Omega, \quad t > 0,\\ & u(x,0) = u_0(x),\quad x\in\bar\Omega, \end{
Sun, Wenjun, Wang, Shu
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Nonlinear Parabolic Equations

1996
We begin this chapter with some general results on the existence and regularity of solutions to semilinear parabolic PDE, first treating the pure initial-value problem in § 1, for PDE of the form $$\frac{\partial u} {\partial t} = Lu + F(t,x,u,\nabla u),\quad u(0) = f,$$ (0.1) where u is defined on [0, T) × M, and M has no boundary.
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Maximum Principle for Nonlinear Parabolic Equations

Journal of Mathematical Sciences, 2018
In this paper, the author gives some maximum principles for solutions of some nonlinear parabolic equations. The author considers first, the parabolic equation \[ \mathcal{L}u-u_{t}=f(x,t,u,Du),\text{ in }\Omega\cup\gamma\Omega\tag{1} \] where {\parindent=6mm \begin{itemize}\item[{\(\bullet\)}] \(\Omega\subset\mathbb{R}^n\times(0,\infty)\) is an ...
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