American Institute of Mathematical Sciences

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Evolution Equations & Control Theory

September 2015 , Volume 4 , Issue 3

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2015, 4(3): 241-263 doi: 10.3934/eect.2015.4.241 +[Abstract](2128) +[PDF](540.7KB)
Abstract:
We analyse the longterm properties of a $C_0-$semigroup describing the solutions to a nonlinear thermoelastic diffusion plate, recently derived by Aouadi [1], where the heat and diffusion flux depends on the past history of the temperature and the chemical potential gradients through memory kernels. First we prove the well-posedness of the initial-boundary-value problem using the $C_0-$semigroup theory of linear operators. Then we show, without rotational inertia, that the thermal and chemical potential coupling is strong enough to guarantee the quasi-stability. By showing that the system is gradient and asymptotically compact, the existence of a global attractor whose fractal dimension is finite is proved.
2015, 4(3): 265-279 doi: 10.3934/eect.2015.4.265 +[Abstract](1910) +[PDF](370.4KB)
Abstract:
In this work we consider the thermal convection problem in arbitrary bounded domains of a three-dimensional space for incompressible viscous fluids, with a fading memory constitutive equation for the heat flux. With the help of a recently proposed free energy, expressed in terms of a minimal state functional for such a system, we prove an existence and uniqueness theorem for the linearized problem. Then, assuming some restrictions on the Rayleigh number, we also prove exponential decay of solutions.
2015, 4(3): 281-296 doi: 10.3934/eect.2015.4.281 +[Abstract](1972) +[PDF](450.4KB)
Abstract:
This paper is addressed to study the null controllability with constraints on the state for the Kuramoto-Sivashinsky equation. We first consider the linearized problem. Then, by Kakutani fixed point theorem, we show that the same result holds for the Kuramoto-Sivashinsky equation.
2015, 4(3): 297-314 doi: 10.3934/eect.2015.4.297 +[Abstract](1599) +[PDF](416.8KB)
Abstract:
We establish a theorem combining the estimates of Ingham and Müntz--Szász. Moreover, we allow complex exponents instead of purely imaginary exponents for the Ingham type part or purely real exponents for the Müntz--Szász part. A very special case of this theorem allows us to prove the simultaneous observability of some string--heat and beam--heat systems.
2015, 4(3): 315-324 doi: 10.3934/eect.2015.4.315 +[Abstract](2327) +[PDF](369.8KB)
Abstract:
In this paper, we consider the Cauchy problem of a sixth order Cahn-Hilliard equation with the inertial term, \begin{eqnarray*} ku_{t t} + u_t - \Delta^3 u - \Delta(-a(u) \Delta u -\frac{a'(u)}2|\nabla u|^2 + f(u))=0. \end{eqnarray*} Based on Green's function method together with energy estimates, we get the global existence and optimal decay rate of solutions.
2015, 4(3): 325-346 doi: 10.3934/eect.2015.4.325 +[Abstract](2675) +[PDF](519.4KB)
Abstract:
In this paper, we study the exact controllability of a second order linear evolution equation in a domain with highly oscillating boundary with homogeneous Neumann boundary condition on the oscillating part of boundary. Our aim is to obtain the exact controllability for the homogenized equation. The limit problem with Neumann condition on the oscillating boundary is different and hence we need to study the exact controllability of this new type of problem. In the process of homogenization, we also study the asymptotic analysis of evolution equation in two setups, namely solution by standard weak formulation and solution by transposition method.
2015, 4(3): 347-353 doi: 10.3934/eect.2015.4.347 +[Abstract](1802) +[PDF](312.1KB)
Abstract:
We consider the wave equation $u_{tt}=\Delta u$ on a bounded domain $\Omega\subset{\mathbb R}^n$, $n>1$, with smooth boundary of positive mean curvature. On the boundary, we impose the absorbing boundary condition ${\partial u\over\partial\nu}+u_t=0$. We prove uniqueness of solutions backward in time.
2015, 4(3): 355-372 doi: 10.3934/eect.2015.4.355 +[Abstract](2323) +[PDF](504.7KB)
Abstract:
We investigate the initial value problems for some semilinear wave, heat and Schrödinger equations in two space dimensions, with exponential nonlinearities. Using the potential well method based on the concepts of invariant sets, we prove either global well-posedness or finite time blow-up.

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