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We present a blow-up result for large data for relaxed compressible Navier-Stokes models avoiding the possibility of reaching the boundary of hyperbolicity. Thus a previous result is improved and further examples are given illustrating possible effects of a relaxation and contrasting the classical compressible Navier-Stokes equations without relaxation where solutions for large data exist globally., Comment: 10 pages

We consider the non-isentropic compressible Navier-Stokes equations in two or three space dimensions for which the heat conduction of Fourier's law is replaced by Cattaneo's law and the classical Newtonian flow is replaced by a revised Maxwell flow. We show that a physical entropy exists for this new model. For two special cases, we show the global well-posedness of solutions with small initial data and the blow-up of solutions in finite time for a class of large initial data. Moreover, for vanishing relaxation parameters, the solutions (if it exists) are shown to converge to solutions of the classical system., Comment: 26 pages

We propose a Hilbert space solution theory for a nonhomogeneous heat equation with delay in the highest order derivatives with nonhomogeneous Dirichlet boundary conditions in a bounded domain. Under rather weak regularity assumptions on the data, we prove a well-posedness result and give an explicit representation of solutions. Further, we prove an exponential decay rate for the energy in the dissipative case. We also show that lower order regularizations lead to ill-posedness, also for higher-order equations. Finally, an application with physically relevant constants is given., Comment: 34 pages, 5 figures

We investigate the dispersive properties of evolution equations on waveguides with a non flat shape. More precisely we consider an operator $H=-\Delta_{x}-\Delta_{y}+V(x,y)$ with Dirichled boundary condition on an unbounded domain $\Omega$, and we introduce the notion of a \emph{repulsive waveguide} along the direction of the first group of variables $x$. If $\Omega$ is a repulsive waveguide, we prove a sharp estimate for the Helmholtz equation $Hu-\lambda u=f$. As consequences we prove smoothing estimates for the Schr\"odinger and wave equations associated to $H$, and Strichartz estimates for the Schr\"odinger equation. Additionally, we deduce that the operator $H$ does not admit eigenvalues., Comment: 22 pages, 4 figures