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Vibrational Control of Nonlinear Parabolic Systems
IFAC Proceedings Volumes, 1987Abstract Vibrational control is a non-classical control principle which proposes utilization of zero mean parametric excitation of a dynamical system to achieve control objectives. The present paper is devoted to the development of vibrational control theory of nonlinear infinite dimensional systems described by parabolic partial differential ...
J. Bentsman, S.M. Meerkov, Xianshu Shu
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Regularity for Certain Nonlinear Parabolic Systems
Communications in Partial Differential Equations, 2004Abstract By means of an inequality of Poincare type, a weak Harnack inequality for the gradient of a solution and an integral inequality of Campanato type, it is shown that a solution to certain degenerate parabolic system is locally Holder continuous. The system is a generalization of p-Laplacian system.
Hyeong-Ohk Bae, Hi Jun Choe
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Optimal Problems for Nonlinear Parabolic Boundary Control Systems
SIAM Journal on Control and Optimization, 1994Some control problems (in particular the time optimal problem) are considered for the heat equation: \[ y_ t (t,x)= \Delta y(t,x), \quad y(0,x)= \zeta (x) \qquad (x\in \Omega ...
Fattorini, H. O., Murphy, T.
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Diffusion-Induced Blowup in a Nonlinear Parabolic System
Journal of Dynamics and Differential Equations, 1998The authors consider the nonlinear diffusion system \[ u_t= d_u\Delta u+ f(u,v),\quad v_t= d_v\Delta v+ g(u,v)\quad\text{in }\Omega\times (0,\infty)\tag{1} \] with Neumann boundary conditions \[ \partial u/\partial\nu(x, t)= \partial v/\partial\nu(x, t)= 0\quad\text{on }\partial\Omega\times (0,\infty)\tag{2} \] and the initial conditions.
Mizoguchi, Noriko +2 more
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1991
We examine existence, uniqueness, and regularity of solutions to nonlinear parabolic systems. We begin with an approach to strongly parabolic quasilinear equations using techniques very similar to those applied to hyperbolic systems in Chapter 5, moving on to symmetrizable quasilinear parabolic systems in §7.2.
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We examine existence, uniqueness, and regularity of solutions to nonlinear parabolic systems. We begin with an approach to strongly parabolic quasilinear equations using techniques very similar to those applied to hyperbolic systems in Chapter 5, moving on to symmetrizable quasilinear parabolic systems in §7.2.
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On parabolic systems with discontinuous nonlinearities
Applicable Analysis, 1985We consider parabolic systems over (O,T)×Ω with bounded but discontinuous nonline-arities. Here A1,A2 are positive elliptic operators of order 2 m with continuous coefficients, f is a bounded function having a jump in u=1, and g1, g2 are Lipschitz continuous and bounded. We prescribe Dirichlet boundary conditions and the initial values u(O)=φ1, v(O)=φ2.
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Boundedness in a parabolic-parabolic chemotaxis system with nonlinear diffusion
Zeitschrift für angewandte Mathematik und Physik, 2013zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Wang, Liangchen +2 more
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Invariant Sets for Nonlinear Elliptic and Parabolic Systems
SIAM Journal on Mathematical Analysis, 1980In this paper we consider systems of weakly coupled nonlinear second order elliptic and parabolic equations with nonlinear, possibly coupled, boundary conditions. The aim is to find invariant sets of the form \[ S = \left\{ {\left( {u_1 ,u_2 , \cdots ,u_m } \right)|\varphi _i (x) \leqq u_i (x) \leqq \psi _i (x){\text{ a.e.}}} \right\}\] for certain ...
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Optimal control of a nonlinear parabolic–elliptic system
Nonlinear Analysis: Theory, Methods & Applications, 2009zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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Asymptotical self-similarity in nonlinear parabolic systems
Nonlinear Differential Equations and Applications NoDEA, 2005We prove that blowing up solutions of the system $$u_{{it}} - d_{i} \Delta u_{i} = {\prod\limits_{k = 1}^m {u_{k} ^{{p_{k} ^{i} }} ,} }\quad i = 1, \ldots ,m,\;x \in \mathbb{R}^{N} ,\;t > 0,$$ $$u_{i} (0,x) = u_{{0i}} (x),\quad i = 1, \ldots ,m,\;x \in \mathbb{R}^{N} ,$$ where m > 1 and pki ≤ 0, behave like self-similar solutions to this ...
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