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KNESER FAMILIES IN SEMILINEAR PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS (Variational Problems and Related Topics)

, 2001
Takashi Kaminogo
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Fourth order semilinear parabolic equations

Fourth order semilinear parabolic equations
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Stabilizing Semilinear Parabolic Equations

Numerical Functional Analysis and Optimization, 2005
In this paper we prove the internal feedback stabilization of steady-state solutions of semilinear parabolic equations and introduce a controller synthesis methodology based on finite element approximations of the original PDEs. Numerical tests are given for some one- and two-dimensional nonlinear parabolic equations.
V. Barbu, D. Coca, Y. Yan
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GLOBAL SOLUTION OF A SEMILINEAR PARABOLIC EQUATION

Acta Mathematica Scientia, 1991
The authors consider the semilinear parabolic equation \(u_ t=\Delta u+| u|^{\gamma-1}u-u\), where \((x,t)\in\mathbb{R}^ n\times\mathbb{R}^ +\) \((\gamma>1)\), in the weighted Lebesgue class \(D_ q^ \alpha=\{u\in L_ q:\;| x|^ \alpha u\in L_ q\}\). Existence and asymptotic behaviour results of the global solutions for the associated Cauchy problem with ...
Ding, Xiaxi, Zhao, Huijiang
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Numerical Methods for Semilinear Parabolic Equations

SIAM Journal on Numerical Analysis, 1987
The parabolic problem: (1) \(u_ t-\nabla \cdot (D(\chi,t)\nabla u)=f(\chi,t,u)\) for \(x\in \Omega\), \(0\leq t\leq T\); \(\alpha (\chi_ 0)\partial u/\partial \nu +\beta (\chi_ 0)u=h(\chi_ 0,t)\) for \(\chi_ 0\in \partial \Omega\), \(0\leq t\leq T\), and \(u(0,\chi)=\psi (\chi)\) for \(\chi\in \Omega\) is approximated in the usual way by an implicit ...
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Lifespan for a semilinear pseudo‐parabolic equation

Mathematical Methods in the Applied Sciences, 2017
This paper deals with the blow‐up solution to the following semilinear pseudo‐parabolic equation urn:x-wiley:mma:media:mma4639:mma4639-math-0001 in a bounded domain , which was studied by Luo (Math Method Appl Sci 38(12):2636‐2641, 2015) with the following assumptions on p: urn:x-wiley:mma:media:mma4639:mma4639-math-0003 and the lifespan for the ...
Guangyu Xu, Jun Zhou
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Generalized Solutions of Semilinear Parabolic Equations

Monatshefte für Mathematik, 2005
The Cauchy problem for the semilinear parabolic equation \(u_t=\Delta_x u+f(u)\) in \(t>0\), \(x\in \mathbb R^n\) is studied in the framework of algebra of generalized functions introduced by Colombeau. By using approximations for generalized functions, the results on existence and uniqueness of generalized functions solution are obtained -- in ...
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Optimal Control of Nonsmooth, Semilinear Parabolic Equations

SIAM Journal on Control and Optimization, 2017
This paper is concerned with an optimal control problem governed by nonsmooth semilinear parabolic equations. The essential feature of the problem is that the nonlinearity in the state equation is only Lipschitz continuous and not necessarily Gateaux-differentiable.
Meyer, Christian, Susu, Livia M.
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W-methods for semilinear parabolic equations

Applied Numerical Mathematics, 1995
The author studies the temporal convergence behavior of \(W\)-methods applied to the initial value problem \[ u'(t)+ Au(t)= g(t, u(t)), \quad u(t_0) \text{ given}, \] in an arbitrary Banach space \(X\). The operator \(A\) is not necessarily bounded. The stability and convergence analysis uses the framework of analytic semigroups of linear operators and
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