Quasilinear Diffusion Equation Research Articles

Stability of the logarithmic Sobolev inequality and uncertainty principle for the Tsallis entropy

We consider the stability of the functional inequalities concerning the entropy functional. For the Boltzmann–Shannon entropy, the logarithmic Sobolev inequality holds as a lower bound of the entropy by the Fisher information, and the Heisenberg uncertainty principle follows from combining it with the Shannon inequality. The optimizer for these inequalities is the Gauss function, which is a fundamental solution to the heat equation. In the fields of statistical mechanics and information theory, the Tsallis entropy is known as a one-parameter extension of the Boltzmann–Shannon entropy, and the Wasserstein gradient flow of it corresponds to the quasilinear diffusion equation. We consider the improvement and stability of the optimizer for the logarithmic Sobolev inequality related to the Tsallis entropy. Furthermore, we show the stability results of the uncertainty principle concerning the Tsallis entropy.

Eternal solutions to a porous medium equation with strong non-homogeneous absorption. Part I: radially non-increasing profiles

Existence of specific eternal solutions in exponential self-similar form to the following quasilinear diffusion equation with strong absorption \[ \partial_t u=\Delta u^m-|x|^{\sigma}u^q, \] posed for $(t,\,x)\in (0,\,\infty )\times \mathbb {R}^N$ , with $m>1$ , $q\in (0,\,1)$ and $\sigma =\sigma _c:=2(1-q)/ (m-1)$ is proved. Looking for radially symmetric solutions of the form \[ u(t,x)={\rm e}^{-\alpha t}f(|x|\,{\rm e}^{\beta t}), \quad \alpha=\frac{2}{m-1}\beta, \] we show that there exists a unique exponent $\beta ^*\in (0,\,\infty )$ for which there exists a one-parameter family $(u_A)_{A>0}$ of solutions with compactly supported and non-increasing profiles $(f_A)_{A>0}$ satisfying $f_A(0)=A$ and $f_A'(0)=0$ . An important feature of these solutions is that they are bounded and do not vanish in finite time, a phenomenon which is known to take place for all non-negative bounded solutions when $\sigma \in (0,\,\sigma _c)$ .

Optimal existence, uniqueness and blow-up for a quasilinear diffusion equation with spatially inhomogeneous reaction

Well-posedness and a number of qualitative properties for solutions to the Cauchy problem for the following nonlinear diffusion equation with a spatially inhomogeneous source∂tu=Δum+|x|σup, posed for (x,t)∈RN×(0,T), with exponents 1<p<m and σ>0, are established. More precisely, we identify the optimal class of initial conditionsu0 for which (local in time) existence is ensured and prove non-existence of solutions for the complementary set of data. We establish then (local in time) uniqueness and a comparison principle for this class of data. We furthermore prove that any non-trivial solution to the Cauchy problem blows up in a finite timeT∈(0,∞) and finite speed of propagation holds true for t∈(0,T): if u0∈L∞(RN) is an initial condition with compact support and blow-up time T>0, then u(t) is compactly supported for t∈(0,T). We also establish in this work the absence of localization at the blow-up time T for solutions stemming from compactly supported data.

An accurate, robust, and efficient weak Galerkin finite element scheme with graded meshes for the time-fractional quasi-linear diffusion equation

A time-fractional quasi-linear diffusion equation with a Caputo time derivative of order 0 < α < 1 is considered. The weak Galerkin finite element method is used for the space discretization, the Caputo time derivative is discretized by L 1 method and Newton linear method is used for the quasi-linear term with graded meshes in the temporal direction. The stability and convergence for fully discrete weak Galerkin finite element scheme in L 2 -norm are proved. Numerical experiments are given to confirm the theoretical results.

Resonance broadening effect for relativistic electron interaction with electromagnetic ion cyclotron waves

Relativistic electron scattering by electromagnetic ion cyclotron (EMIC) waves is one of the most effective mechanisms for &amp;gt;1 MeV electron flux depletion in the Earth's radiation belts. Resonant electron interaction with EMIC waves is traditionally described by quasi-linear diffusion equations, although spacecraft observations often report EMIC waves with intensities sufficiently large to trigger nonlinear resonant interaction with electrons. An important consequence of such nonlinear interaction is the resonance broadening effect due to high wave amplitudes. In this study, we quantify this resonance broadening effect in electron pitch-angle diffusion rates. We show that resonance broadening can significantly increase the pitch-angle range of EMIC-scattered electrons. This increase is especially important for ∼1 MeV electrons, where, without the resonance broadening, only those near the loss cone (with low fluxes) can resonate with EMIC waves.

High-order bicompact schemes for the quasilinear multidimensional diffusion equation

High-order bicompact schemes are designed for the quasilinear multidimensional diffusion equation. They are constructed using the method of lines, the finite volume method, the bicubic (or tricubic) Hermite interpolation. Implicit-explicit and diagonally-implicit Runge-Kutta methods are applied to time integration. The resulting schemes are stable for any ratios between grid steps, are conservative, have approximation of the fourth order in space and the third order in time. To implement the implicit-explicit schemes, an iteration method is proposed based on the approximate factorization of their multidimensional difference operators. This method is modified to implement the schemes with diagonally-implicit Runge-Kutta time stepping. High-order grid convergence and implementation efficiency of the new bicompact schemes are demonstrated on numerical examples.

Kinetic maximal Lp-regularity with temporal weights and application to quasilinear kinetic diffusion equations

We introduce the concept of kinetic maximal Lp-regularity with temporal weights and prove that this property is satisfied for the (fractional) Kolmogorov equation. We show that solutions are continuous with values in the trace space and prove, in particular, that the trace space can be characterized in terms of anisotropic Besov spaces. We further extend the property of kinetic maximal Lμp-regularity to the Kolmogorov equation with variable coefficients. Finally, we show how kinetic maximal Lμp-regularity can be used to obtain local existence of solutions to a class of quasilinear kinetic equations and illustrate our result with a quasilinear kinetic diffusion equation.

A gradient maximum principle of solutions for a quasilinear parabolic equation

This short note is concerned with a quasilinear diffusion equation under initial and Neumann boundary value condition. To be more precise, the authors establish a gradient maximum principle of classical solutions via the maximum principle and Hopf’s lemma. The result generalizes a recent work obtained by Kim (Proc Amer Math Soc 145:1203–1208, 2017).

A Quasi-linear Diffusion Model for Resonant Wave–Particle Instability in Homogeneous Plasma

Abstract In this paper, we develop a model to describe the generalized wave–particle instability in a quasi-neutral plasma. We analyze the quasi-linear diffusion equation for particles by expressing an arbitrary unstable and resonant wave mode as a Gaussian wave packet, allowing for an arbitrary direction of propagation with respect to the background magnetic field. We show that the localized energy density of the Gaussian wave packet determines the velocity-space range in which the dominant wave–particle instability and counteracting damping contributions are effective. Moreover, we derive a relation describing the diffusive trajectories of resonant particles in velocity space under the action of such an interplay between the wave–particle instability and damping. For the numerical computation of our theoretical model, we develop a mathematical approach based on the Crank–Nicolson scheme to solve the full quasi-linear diffusion equation. Our numerical analysis solves the time evolution of the velocity distribution function under the action of a dominant wave–particle instability and counteracting damping and shows a good agreement with our theoretical description. As an application, we use our model to study the oblique fast-magnetosonic/whistler instability, which is proposed as a scattering mechanism for strahl electrons in the solar wind. In addition, we numerically solve the full Fokker–Planck equation to compute the time evolution of the electron-strahl distribution function under the action of Coulomb collisions with core electrons and protons after the collisionless action of the oblique fast-magnetosonic/whistler instability.

Optimization of the Matrix Fourier-Filter for a Class of Nonlinear Optical Models with an Integral Objective Functional

We consider a new formulation of the Fourier-filtering problem that uses matrix Fourier-filters as the controls in nonlinear optical models described by quasi-linear functional-differential diffusion equations. Solvability of the control problem is proved for various classes of matrix Fourier-filters with a time-integral objective functional. Differentiability of the functional with respect to the matrix Fourier-filter and convergence of a variant of the gradient projection method are proved. Examples of numerical simulation of controlled structure formation are presented, and the advantages of matrix Fourier-filters compared with traditional multiplier filters are demonstrated.

Numerical Simulation of the Absorption of Laser Radiation in a Plasma Including the Resonance Mechanism and Generation of Fast Electrons for the Conditions of a Nuclear Fusion Target Ignition

A model of resonance absorption of laser radiation in a plasma involving the deformation of the density profile by the ponderomotive force has been proposed. A mechanism of the linear transformation of laser radiation into plasma waves at resonance absorption has been discussed. The quasilinear diffusion equation for the electron distribution function in the plasma field has been used to describe the generation of fast electrons by plasma waves. Formulas have been obtained to estimate the energy fraction absorbed through the resonance mechanism and the energy of fast electrons. These formulas have been used in the two-dimensional hydrodynamic code ATLANT-HE. The hydrodynamics of the plasma and the absorption of laser radiation have been calculated for conditions of the NIF facility experiment at a radiation power of 30 ТW and a base pulse duration of 7.5 ns. The plasma has the spatial dimensions appearing upon the irradiation of a spherical target for the initiation of the fusion reaction. The calculations have shown that the resonance absorption mechanism leads to the generation of fast electrons with an energy of about 40–80 keV. Fast electrons carry an energy of about 1–1.5% of the laser energy. These results are close to the experimental data.

Conditional Lie–Bäcklund Symmetries and Functionally Generalized Separation of Variables to Quasi-Linear Diffusion Equations with Source

The conditional Lie–Bäcklund symmetry method is applied to investigate the functionally generalized separation of variables for quasi-linear diffusion equations with a source. The equations and the admitted conditional Lie–Bäcklund symmetries related to invariant subspaces are identified. The exact solutions possessing the form of the functionally generalized separation of variables are constructed for the resulting equations due to the corresponding symmetry reductions.

On classes of well-posedness for quasilinear diffusion equations in the whole space

<p style="text-indent:20px;">Well-posedness classes for degenerate elliptic problems in <inline-formula><tex-math id="M1">\begin{document}$ {\mathbb R}^N $\end{document}</tex-math></inline-formula> under the form <inline-formula><tex-math id="M2">\begin{document}$ u = \Delta {{\varphi}}(x,u)+f(x) $\end{document}</tex-math></inline-formula>, with locally (in <inline-formula><tex-math id="M3">\begin{document}$ u $\end{document}</tex-math></inline-formula>) uniformly continuous nonlinearities, are explored. While we are particularly interested in the <inline-formula><tex-math id="M4">\begin{document}$ L^\infty $\end{document}</tex-math></inline-formula> setting, we also investigate about solutions in <inline-formula><tex-math id="M5">\begin{document}$ L^1_{loc} $\end{document}</tex-math></inline-formula> and in weighted <inline-formula><tex-math id="M6">\begin{document}$ L^1 $\end{document}</tex-math></inline-formula> spaces. We give some sufficient conditions in order that the uniqueness and comparison properties hold for the associated solutions; these conditions are expressed in terms of the moduli of continuity of <inline-formula><tex-math id="M7">\begin{document}$ u\mapsto {{\varphi}}(x,u) $\end{document}</tex-math></inline-formula>. Under additional restrictions on the dependency of <inline-formula><tex-math id="M8">\begin{document}$ {{\varphi}} $\end{document}</tex-math></inline-formula> on <inline-formula><tex-math id="M9">\begin{document}$ x $\end{document}</tex-math></inline-formula>, we deduce the existence results for the corresponding classes of solutions and data. Moreover, continuous dependence results follow readily from the existence claim and the comparison property. In particular, we show that for a general continuous non-decreasing nonlinearity <inline-formula><tex-math id="M10">\begin{document}$ {{\varphi}}: {\mathbb R}\mapsto {\mathbb R} $\end{document}</tex-math></inline-formula>, the space <inline-formula><tex-math id="M11">\begin{document}$ L^\infty $\end{document}</tex-math></inline-formula> (endowed with the <inline-formula><tex-math id="M12">\begin{document}$ L^1_{loc} $\end{document}</tex-math></inline-formula> topology) is a well-posedness class for the problem <inline-formula><tex-math id="M13">\begin{document}$ u = \Delta {{\varphi}}(u)+f(x) $\end{document}</tex-math></inline-formula>.

Uniform Boundedness and Global Existence of Solutions to a Quasilinear Diffusion Equation with Nonlocal Fisher-KPP Type Reaction Term

This paper deals with the Cauchy problem and Neumann initial boundary value problem for a quasilinear diffusion equation with nonlocal Fisher-KPP type reaction terms. We establish the uniform boundedness and global existence of solutions to the problems by using multipliers technique and modified Moser's iteration argument for some ranges of parameters. Moreover, the ranges of parameters have similar structure to that of the classical critical Fujita exponent.

Drug release enhanced by temperature: An accurate discrete model for solutions in [formula omitted

In this paper we consider the coupling between two quasilinear diffusion equations: the diffusion coefficient of the first equation depends on its solution and the diffusion and convective coefficients of the second equation depend on the solution of the first one. This system can be used to describe the drug evolution in a target tissue when the drug transport is enhanced by heat. We study, from an analytical and a numerical viewpoints, the coupling of the heat equation with the drug diffusion equation. A fully discrete piecewise linear finite method is proposed to solve this system and its stability is studied. Assuming that the heat and the concentration are in H3 we prove that the method is second order convergent. Numerical experiments illustrating the theoretical results and the global qualitative behaviour of the coupling are also included.

Hopf bifurcation in diffusive model of nonlinear optical system with matrix fourier filtering

We consider a new formulation of the Fourier filtering problem based on the use of matrix filters instead of conventional filters-multiplicators in the framework of the periodic boundary value problem for a quasilinear diffusion equation of nonlinear optics. Our attention is mostly focused on studying the capabilities of matrix Fourier filtering as an efficient tool for constructing periodic solutions with prescribed spatial-temporal dynamics. To this end, we analyze the spectrum of the linearized operator and the possibility to localize it by choosing a matrix Fourier filter that provides the existence of the Hopf bifurcation. We also describe the structure of bifurcation solution for two important classes of filters: finite filters-multiplicators and matrix filters. We theoretically and numerically show that along with rotating waves, typical for the Hopf bifurcation with conventional filters-multiplicators, there exist waves with new nontrivial dynamics (rotating waves with nodes and pulsating structures). These waves excite as a result of the Hopf bifurcation under an appropriate choice of nondiagonal matrix Fourier filters.

Non-local Diffusion Equations Involving the Fractional $$p(\cdot )$$-Laplacian

In this paper we study a class of nonlinear quasi-linear diffusion equations involving the fractional $$p(\cdot )$$-Laplacian with variable exponents, which is a fractional version of the nonhomogeneous $$p(\cdot )$$-Laplace operator. The paper is divided into two parts. In the first part, under suitable conditions on the nonlinearity f, we analyze the problem $$({\mathscr {P}}_{1})$$ in a bounded domain $$\varOmega $$ of $${\mathbb {R}}^N$$ and we establish the well-posedness of solutions by using techniques of monotone operators. We also study the large-time behaviour and extinction of solutions and we prove that the fractional $$p(\cdot )$$-Laplacian operator generates a (nonlinear) submarkovian semigroup on $$L^{2}(\varOmega ).$$ In the second part of the paper we establish the existence of global attractors for problem $$({\mathscr {P}}_{2})$$ under certain conditions in the potential $${\mathbb {V}}.$$ Our results are new in the literature, both for the case of variable exponents and for the fractional p-laplacian case with constant exponent.

Well-posedness of quasilinear singular diffusion equations with boundary degeneracy

This paper concerns the well-posedness problems of a class ofquasilinear singular diffusion equations with boundary degeneracy.In addition to the possibledegeneracy and singularity likethe non-Newtonian filtration equations, the equations aredegenerate on a portion of the boundary due to the diffusion coefficients.The diffusion coefficients are anisotropic.For the well-posedness problems of such equations,the boundary condition may not be prescribed on the entire lateral boundary.The prescription of the boundary condition is determined bythe degeneracy of the diffusion coefficients along the outernormal direction of the spatial domain.The well-posedness problems of the equations are formulatedand the existence and uniqueness of weak solutions to the problems are proved.

Rotating waves in a spatially nonlocal delayed feedback optical system with diffraction

We study self-organization in a nonlinear optical system with a feedback loop, where light is delayed and undergoes a spatial transformation within the thin annulus aperture. The dynamics of the system are modelled by a quasilinear functional differential diffusion equation on a circle that tracks the effects of diffraction by means of a coupled linear Schrodinger equation. We prove the existence of rotating wave solutions and study their stability by constructing the Faria normal form of a Hopf bifurcation.

A nonlocal problem with a free boundary for the quasilinear diffusion equation

A nonlocal problem with a free boundary for the quasilinear diffusion equation