eISSN:

2163-2480

## Evolution Equations & Control Theory

June 2016 , Volume 5 , Issue 2

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2016, 5(2): 201-224
doi: 10.3934/eect.2016001

*+*[Abstract](2032)*+*[PDF](507.8KB)**Abstract:**

The problem of stabilization and controllability for inhomogeneous multidimensional thermoelastic diffusion problem is considered for anisotropic material. By introducing a nonlinear feedback function on part of the boundary of the material, which is clamped along the rest of its boundary, we prove that the energy of the system decays to zero exponentially or polynomially. Both rates of decay are determined explicitly by the physical parameters. Via Russell's ``Controllability via Stabilizability" principle, we prove that the considered system is partially controllable by a boundary function determined explicitly.

2016, 5(2): 225-234
doi: 10.3934/eect.2016002

*+*[Abstract](2263)*+*[PDF](389.7KB)**Abstract:**

We consider the Cauchy problem for a nonlinear Dirac equation on $\mathbb{R}^{n}$, $n\ge1$, with a power type,

*non*gauge invariant nonlinearity $\sim|u|^{p}$. We prove several ill-posedness and blowup results for both large and small $H^{s}$ data. In particular we prove that: for (essentially arbitrary) large data in $H^{\frac n2+}(\mathbb{R} ^n)$ the solution blows up in a finite time; for suitable large $H^{s}(\mathbb{R} ^n)$ data and $s< \frac{n}{2}-\frac{1}{p-1}$ no weak solution exist; when $1< p <1+\frac1n$ (or $1< p <1+\frac2n$ in $n=1,2,3$), there exist arbitrarily small initial data data for which the solution blows up in a finite time.

2016, 5(2): 235-250
doi: 10.3934/eect.2016003

*+*[Abstract](2288)*+*[PDF](369.4KB)**Abstract:**

A coupled system of hyperbolic equations in Maxwell/wave with Wentzell conditions in a bounded domain of $\mathbb{R}^3$ is considered. Under suitable assumptions, we show the exponential stability of the system. Our method is based on an identity with multipliers that allows to show an appropriate stability estimate.

2016, 5(2): 251-272
doi: 10.3934/eect.2016004

*+*[Abstract](3016)*+*[PDF](565.4KB)**Abstract:**

We propose a novel PDE-based anisotropic filter for noise reduction in multicolor images. It is a generalization of Nitzberg & Shiota's (1992) model being a hyperbolic relaxation of the well-known parabolic Perona & Malik's filter (1990). First, we consider a `spatial' mollifier-type regularization of our PDE system and exploit the maximal $L^{2}$-regularity theory for non-autonomous forms to prove a well-posedness result both in weak and strong settings. Again, using the maximal $L^{2}$-regularity theory and Schauder's fixed point theorem, respective solutions for the original quasilinear problem are obtained and the uniqueness of solutions with a bounded gradient is proved. Finally, the long-time behavior of our model is studied.

2016, 5(2): 273-302
doi: 10.3934/eect.2016005

*+*[Abstract](2321)*+*[PDF](512.4KB)**Abstract:**

In this work we establish the local solvability of quasilinearsymmetric hyperbolic system using local monotonicity method andfrequency truncation method. The existence of an optimal control isalso proved as an application of these methods.

2016, 5(2): 303-335
doi: 10.3934/eect.2016006

*+*[Abstract](1871)*+*[PDF](618.2KB)**Abstract:**

In this article, we set up the continuous maximal regularity theory for a class of linear differential operators on manifolds with singularities. These operators exhibit degenerate or singular behaviors while approaching the singular ends. Particular examples of such operators include differential operators defined on domains, which degenerate fast enough toward the boundary. Applications of the theory established herein are shown to the Yamabe flow, the porous medium equation, the parabolic $p$-Laplacian equation and the thin film equation. Some comments about the boundary blow-up problem, and waiting time phenomenon for singular or degenerate parabolic equations can also be found in this paper.

2016, 5(2): 337-348
doi: 10.3934/eect.2016007

*+*[Abstract](1994)*+*[PDF](372.7KB)**Abstract:**

In this paper, we study the initial value problem for the three-dimensional nematic liquid crystal flows. Blow up criterion of smooth solutions is established by the energy method, which refines the previous result.

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