Degenerate Diffusion Equation Research Articles

Wavefronts and global stability in a time-delayed population model with stage structure

We formulate and study a one–dimensional single–species diffusive–delay population model. The time delay is the time taken from birth to maturity. Without diffusion, the delay differential model extends the well–known logistic differential equation by allowing delayed constant birth processes and instantaneous quadratically regulated death processes. This delayed model is known to have simple global dynamics similar to that of the logistic equation. Through the use of a sub/supersolution pair method, we show that the diffusive delay model continues to generate simple global dynamics. This has the important biological implication that quadratically regulated death processes dramatically simplify the growth dynamics. We also consider the possibility of travelling wavefront solutions of the scalar equation for the mature population, connecting the zero solution of that equation with the positive steady state. Our main finding here is that our fronts appear to be all monotone, regardless of the size of the delay. This is in sharp contrast to the frequently reported findings that delay causes a loss of monotonicity, with the front developing a prominent hump in some other delay models.

Blow-up for degenerate parabolic equations with nonlocal source

This paper deals with the blow-up properties of the solution to the degenerate nonlinear reaction diffusion equation with nonlocal source $x^{q}u_{t}-(x^{\gamma }u_{x})_{x}=\int _{0}^{a}u^{p}dx$ in $(0,a)\times (0,T)$ subject to the homogeneous Dirichlet boundary conditions. The existence of a unique classical nonnegative solution is established and the sufficient conditions for the solution exists globally or blows up in finite time are obtained. Furthermore, it is proved that under certain conditions the blow-up set of the solution is the whole domain.

Semi-implicit complementary volume scheme for solving level set like equations in image processing and curve evolution

We introduce linear semi-implicit complementary volume numerical scheme for solving level set like nonlinear degenerate diffusion equations arising in image processing and curve evolution problems. We study discretization of image selective smoothing equation of mean curvature flow type given by Alvarez, Lions and Morel ([3]). Solution of the level set equation of Osher and Sethian ([26], \[30]) is also included in the study. We prove \(L_\infty\) and \(W^{1,1}\) estimates for the proposed scheme and give existence of its (generalized) solution in every discrete time-scale step. Efficiency of the scheme is given by its linearity and stability. Preconditioned iterative solvers are used for computing arising linear systems. We present computational results related to image processing and plane curve evolution.

Numerical approach to the waiting time for the one-dimensional porous medium equation

We consider the nonlinear degenerate diffusion equation. The most striking manifestation of the nonlinearity and degeneracy is an appearance of interfaces. Under some condition imposed on the initial function, the interfaces do not move on some time interval [ 0 , t ∗ ] \left [ {0,{t^*}} \right ] . In this paper, from numerical points of view, we try to determine the value of t ∗ {t^*} , which is called the waiting time.

Intermediate asymptotics for Richards' equation in a finite layer

Perturbation methods are applied to study an initial-boundary-value problem for Richards' equation, describing vertical infiltration of water into a finite layer of soil. This problem for the degenerate diffusion equation with convection and Dirichlet/Robin boundary conditions exhibits several different regimes of behavior. Boundary-layer analysis and short-time asymptotics are used to describe the structure of similarity solutions, traveling waves, and other solution states and the transitions connecting these different intermediate asymptotic regimes.

Variational Numerical Methods for Solving Nonlinear Diffusion Equations Arising in Image Processing

In this paper we give a general, robust, and efficient approach for numerical solutions of partial differential equations (PDEs) arising in image processing and computer vision. The well-established variational computational techniques, namely, finite element, finite volume, and complementary volume methods, are introduced on a common base to solve nonlinear problems in image multiscale analysis. Since they are based on principles like minimization of energy (finite element method) or conservation laws (finite and complemetary volume methods), they have strong physical backgrounds. They allow clear and physically meaningful derivation of difference equations that are local and easy to implement. The variational methods are combined with semi-implicit discretization in scale, which gives favorable stability and efficiency properties of computations. We show here L∞-stability without any restrictions on scale steps. Our approach leads finally to solving linear systems in every discrete scale level, which can be done efficiently by fast preconditioned iterative solvers. We discuss such computational schemes for the regularized (in the sense of F. Catté et al., SIAM J. Numer. Anal.129, 1992, 182–193) Perona–Malik anisotropic diffusion equation (P. Perona and J. Malik, IEEE Trans. Pattern Anal. Mach. Intell.12, 1990, 629–639) and for nonlinear degenerate diffusion equation of mean curvature flow type studied by L. Alvarez et al. (SIAM J. Numer. Anal.129, 1992, 845–866).

Time rescaling and asymptotic behavior of some fourth-order degenerate diffusion equations

In this paper, we investigate the large-times behavior of weak solutions to the fourth-order degenerate parabolic equation u t = −(| u| n u xxx ) x modeling the evolution of thin films. In particular, for all n > 0, we prove exponential decay of u( x, t) towards its mean value (1/|Ω|) ∫ Ω u dx in L 1-norm for long times and we give the explicit ( n-dependent) rate of decay. The result is based on classical entropy estimates, and on detailed lower bounds for the entropy production.

An existence and uniqueness result for a phase-field model of diffusion-induced grain-boundary motion

We consider a phase-field model for diffusion-induced grain boundary motion. The model couples a parabolic variational inequality to a degenerate diffusion equation. Using a regularization technique, we prove an existence theorem for the resulting system. We also obtain a uniqueness result, provided the solution has some additional regularity.

Self-Similarity and Beyond: Exact Solutions of Nonlinear Problems

<i>Self-Similarity and Beyond: Exact Solutions of Nonlinear Problems</i>

An improperly posed problem for a degenerate nonlinear diffusion equation

An improperly posed problem for a degenerate nonlinear diffusion equation

Asymptotic behaviour of the thin film equation in bounded domains

We investigate the extinction behaviour of a fourth order degenerate diffusion equation in a bounded domain, the model representing the flow of a viscous fluid over edges at which zero contact angle conditions hold. The extinction time may be finite or infinite and we distinguish between the two cases by identification of appropriate similarity solutions. In certain cases, an unphysical mass increase may occur for early time and the solution may become negative; an appropriate remedy for this is noted. Numerical simulations supporting the analysis are included.

Travelling waves in a nonlinear degenerate diffusion model for bacterial pattern formation

We study a reaction diffusion model recently proposed in [5] to describe the spatiotemporal evolution of the bacterium <em>Bacillus subtilis</em> on agar plates containing nutrient. An interesting mathematical feature of the model, which is a coupled pair of partial differential equations, is that the bacterial density satisfies a degenerate nonlinear diffusion equation. It was shown numerically that this model can exhibit quasi-one-dimensional constant speed travelling wave solutions. We present an analytic study of the existence and uniqueness problem for constant speed travelling wave solutions. We find that such solutions exist only for speeds greater than some threshold speed giving minimum speed waves which have a sharp profile. For speeds greater than this minimum speed the waves are smooth. We also characterise the dependence of the wave profile on the decay of the front of the initial perturbation in bacterial density. An investigation of the partial differential equation problem establishes, via a global existence and uniqueness argument, that these waves are the only long time solutions supported by the problem. Numerical solutions of the partial differential equation problem are presented and they confirm the results of the analysis.

A nonlinear degenerate diffusion equation not in divergence form

We consider positive solution of the nonlinear degenerate diffusion equation \(u_t=u^p(\Delta u+u)\) with Dirichlet boundary condition and \(p>1\). It is proved that all positive solutions exist globally if and only if \(\lambda_1\ge 1\), where \(\lambda _1\) is the first eigenvalue of \(-\Delta\) on \(\Omega\) with homogeneous Dirichlet boundary condition.

Large time behaviour of solutions of scalar viscous and nonviscous conservation laws

We study the large time behaviour of entropy solutions of the Cauchy problem for a possibly degenerate nonlinear diffusion equation with a nonlinear convection term. The initial function is assumed to have bounded total variation. We prove the convergence of the solution to the entropy solution of a Riemann problem for the corresponding first-order conservation law.

Positivity-Preserving Numerical Schemes for Lubrication-Type Equations

Lubrication equations are fourth order degenerate diffusion equations of the form $h_t + \nabla \cdot (f(h) \nabla \Delta h) = 0$, describing thin films or liquid layers driven by surface tension. Recent studies of singularities in which $h\to 0$ at a point, describing rupture of the fluid layer, show that such equations exhibit complex dynamics which can be difficult to simulate accurately. In particular, one must ensure that the numerical approximation of the interface does not show a false premature rupture. Generic finite difference schemes have the potential to manifest such instabilities especially when underresolved. We present new numerical methods, in one and two space dimensions, that preserve positivity of the solution, regardless of the spatial resolution, whenever the PDE has such a property. We also show that the schemes can preserve positivity even when the PDE itself is only known to be nonnegativity preserving. We prove that positivity-preserving finite difference schemes have unique positive solutions at all times. We prove stability and convergence of both positivity-preserving and generic methods, in one and two space dimensions, to positive solutions of the PDE, showing that the generic methods also preserve positivity and have global solutions for sufficiently fine meshes. We generalize the positivity-preserving property to a finite element framework and show, via concrete examples, how this leads to the design of other positivity-preserving schemes.

Segregation and mixing in degenerate diffusion in population dynamics

We study the qualitative properties of degenerate diffusion equations used to describe dispersal processes in population dynamics. For systems of interacting populations, the forms of the diffusion models used determine if the population will intermix or remain disjoint (segregated). The dynamics and stability of segregation boundaries between competing populations is analyzed. General population models with segregation and mixing interactions are derived and connections to behavior in fluid mechanical systems are addressed.

A degenerate diffusion equation with a nonlinear source term

A degenerate diffusion equation with a nonlinear source term

Global Solvability of Nonlinear Diffusion Equations with Forcing at the Boundary

Under the influence of a sufficiently “weak” nonlinear source term, it is by now well known that a degenerate diffusion equation is globally solvable. A similar result is known when the nonlinear source is present as a forcing term at the boundary. Such results are usually established via comparison with solutions of a related differential equation or some other form of the maximum principle. However, these techniques do not appear to apply in general situations when forcing occurs on only part of the boundary and convection is present. In this work, we obtain the global existence of solutions for such problems by deriving a differential inequality involving theLr+1norms of solutions. The result is similar to that described above for equations with a nonlinear source term and, furthermore, establishes the global existence of “small” solutions even for “strong” forcings. The technique also applies to a more general class of problems involving nonlinear reaction, diffusion, and convection.

Symmetric Singularity Formation in Lubrication-Type Equations for Interface Motion

Fourth-order degenerate diffusion equations arise in a “lubrication approximation” of a thin film or neck driven by surface tension. Numerical studies of the lubrication equation (LE) $h_t + ( h^n h_{xxx} )_x = 0$ with various boundary conditions indicate that singularity formation in which $h( x( t ),t ) \to 0$ occurs for small enough n with “anomalous” or “second type” scaling inconsistent with usual dimensional analysis. This paper considers locally symmetric or even singularities in the (LE) and in the modified lubrication equation (MLE) $h_t + h^n h_{xxxx} = 0$. Both equations have the property that entropy bounds forbid finite-time singularities when n is sufficiently large. Power series expansions for local symmetric similarity solutions are proposed for equation (LE) with $n < 1$ and (MLE) for all $n \in \mathbb{R}$. In the latter case, special boundary conditions that force singularity formation are required to produce singularities when n is large. Matching conditions at higher-order terms in the expansion suggest a simple functional form for the time dependence of the solution. Computer simulations presented here resolve the self similarity in the onset of the singularity for approximately 30 decades in $\min _x ( {h( {x,t} )} )$. Measurements of the similarity shape and time dependences show excellent agreement with the theoretical prediction. One striking feature of the solution to (MLE) is a transition from a finite-time singularity to an infinite-time singularity at $n = 3/2$. Also, both equations (LE) and (MLE) exhibit symmetric singularities for $n < 0$ with derivatives of order k vanishing at the singular point for all $2 < k < 4 - 2n$. They also exhibit a blowup in derivatives of order greater than $4 - 2n$.

The lubrication approximation for thin viscous films: Regularity and long-time behavior of weak solutions

We consider the fourth-order degenerate diffusion equation, in one space dimension. This equation, derived from a lubrication approximation, models the surface-tension-dominated motion of thin viscous films and spreading droplets [15]. The equation with f(h) = |h| also models a thin neck of fluid in the Hele-Shaw cell [10], [11], [23]. In such problems h(x,t) is the local thickness of the the film or neck. This paper considers the properties of weak solutions that are more relevant to the droplet problem than to Hele-Shaw. For simplicity we consider periodic boundary conditions with the interpretation of modeling a periodic array of droplets. We consider two problems: The first has initial data h0 ≥ 0 and f(h) = |h|n, 0 < n < 3. We show that there exists a weak nonnegative solution for all time. Also, we show that this solution becomes a strong positive solution after some finite time T*, and asymptotically approaches its means as t ∞. The weak solution is in the classical sense of distributions for 3/8 < n < 3 and in a weaker sense introduced in [1] for the remaining 0 < n ≤ 3/8. Furthermore, the solutions have high enough regularity to just include the unique source-type solutions [2] with zero slope at the edge of the support. They do not include any of the less regular solutions with positive slope at the edge of the support. Second, we consider strictly positive initial data h0 ≥ m > 0 and f(h) = |h|n, 0 < n < ∞. For this problem we show that even if a finite-time singularity of the form h 0 does occur, there exists a weak nonnegative solution for all time t. This weak solution becomes strong and positive again after some critical time T*. As in the first problem, we show that the solution approaches its mean as t ∞. The main technical idea is to introduce new classes of dissipative entropies to prove existence and higher regularity. We show that these entropies are related to norms of the difference between the solution and its mean to prove the relaxation result. © 1996 John Wiley & Sons, Inc.