We prove a differential Harnack inequality for the Endangered Species Equation, which is a nonlinear parabolic equation. Our derivation relies on an idea related to the parabolic maximum principle. As an application of this inequality, we will show that positive solutions to this equation must blowup in finite time.

A free boundary problem for the dynamics of a glasslike binary fluid naturally leads to singular perturbation strongly degenerate parabolic partial differential equation in 1D. We present conjecture an asymptotic formula velocity and prove weak version conjecture. The results are based on analysis family local travelling wave solutions.

We show an efficient mechanism to control optical turbulence in systems with different dispersion laws, including parabolic, sub-diffractive, hyper-diffractive or general fractional dispersion. The method is based on the modification of energy cascade through spatial scales leading turbulence: a non-Hermitian spatio-temporal periodic potential allows unidirectional coupling between modes excita...

The paper suggests a modification of the contracting mapping method for non-linear and non-local parabolic equations. This modification is based on weighted in time energy estimates for the L2-norm of the solution of a parabolic equation via a weighted version of the H−1-norm of the free term such that the inverse matrix of the higher order coefficients of the parabolic equation is included int...

On the rectangular space-time domain G = D (0; T ] with the boundary S = GnG, where D = (0; d), we consider the Dirichlet problem for the singularly perturbed parabolic equation: " 2 @ @x a(x; t) @ @x u(x; t) ! ? c(x; t)u(x; t) ? p(x; t) @ @t u(x; t) = f(x; t); (1a) The perturbation parameter " may take any small positive value, say " 2 (0; 1]. If " = 0, the parabolic equation (1a) degenerates ...

We study the asymptotic behavior of solutions to wave equations with structural damping term u t ? ? + 2 = 0 , ( x ) 1 in whole space. New thresholds are reported this paper that indicate which diffusion property and non-diffusive structure dominates low regularity cases. develop end previous authors' research [2] where they have proposed a threshold expresses whether parabolic-like or wave-lik...

This paper deals with works on radar links in a three-dimensional marine environment. Radar performances prediction represents an important phase in the radar system parameterisation. This prediction is strongly related to the electromagnetic waves propagation modelling in maritime environment. The modelling of propagation, in troposphere, requires a good consideration of meteorological paramet...

Abstract We consider a degenerate scalar parabolic equation, in one spatial dimension, of flux-saturated type. The equation also contains convective term. study the existence and regularity traveling-wave solutions; particular we show that they can be discontinuous. Uniqueness is recovered by requiring an entropy condition, entropic solutions turn out to vanishing-diffusion limits with addition...