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This paper is concerned with the critical sharp traveling wave for doubly nonlinear diffusion equation with time delay, where the doubly nonlinear degenerate diffusion is defined by $\Big(\big|(u^m)_x\big|^{p-2}(u^m)_x\Big)_x$ with $m>0$ and $p>1$. The doubly nonlinear diffusion equation is proved to admit a unique sharp type traveling wave for the degenerate case $m(p-1)>1$, the so-called slow-diffusion case. This sharp traveling wave associated with the minimal wave speed $c^*(m,p,r)$ is monotonically increasing, where the minimal wave speed satisfies $c^*(m,p,r)0$. The sharp front is $C^1$-smooth for $\frac{1}{p-1}, Comment: arXiv admin note: text overlap with arXiv:1909.11751

This paper is concerned with a class of nonlocal reaction-diffusion equations with time-delay and degenerate diffusion. Affected by the degeneracy of diffusion, it is proved that, the Cauchy problem of the equation possesses the Holder-continuous solution. Furthermore, the non-critical traveling waves are proved to be globally L 1 -stable, which is the first frame work on L 1 -wavefront-stability for the degenerate diffusion equations. The time-exponential convergence rate is also derived. The adopted approach for the proof is the technical L 1 -weighted energy estimates combining the compactness analysis, but with some new development.

The initial value problems of bipolar isentropic/non-isentropic compressible Navier-Stokes-Maxwell (CNS-M) systems arising from plasmas in R 3 are studied. The main difficulty of studying the bipolar isentropic/non-isentropic CNS-M systems lies in the appearance of the electromagnetic fields satisfying the hyperbolic Maxwell equations. The large time-decay rates of global smooth solutions with small amplitude in L q ( R 3 ) for 2 ≤ q ≤ ∞ are established. For the bipolar non-isentropic CNS-M system, the difference of velocities of two charged carriers decay at the rate ( 1 + t ) − 3 4 + 1 4 q which is faster than the rate ( 1 + t ) − 3 4 + 1 4 q ( ln ⁡ ( 3 + t ) ) 1 − 2 q of the bipolar isentropic CNS-M system, meanwhile, the magnetic field decay at the rate ( 1 + t ) − 3 4 + 3 4 q ( ln ⁡ ( 3 + t ) ) 1 − 2 q which is slower than the rate ( 1 + t ) − 3 4 + 3 4 q for the bipolar isentropic CNS-M system. The approach adopted is the classical energy method but with some new developments, where the techniques of choosing symmetrizers and the spectrum analysis on the linearized homogeneous system play the crucial roles.

The existence of stationary subsonic solutions and their stability for 3-D hydrodynamic model of unipolar semiconductors with the Ohmic contact boundary have been open for long time due to some technical reason, as we know. In this paper, we consider 3-D radial solutions to the system in a hollow ball, and prove that the 3-D radial subsonic stationary solutions uniquely exist and are asymptotically stable, when the initial perturbations around the subsonic steady-state are small enough. Different from the existing studies on the radial solutions for fluid dynamics where the inner boundary of the hollow ball must be far away from the singular origin, here we may allow the chosen inner boundary arbitrarily close to the singular origin and reveal the relationship between the inner boundary and the large time behavior of the radial solution. This partially answers the open question of the stability of stationary waves subjected to the Ohmic contact boundary conditions in the multiple dimensional space. We also prove the existence of non-flat stationary subsonic solution, which essentially improve and develop the previous studies in this subject. The proof is based on the technical energy estimates in certain weighted Sobolev spaces, where the weight functions are artfully selected to be the distance of the targeted spatial location and the singular point.

In this paper, we investigate the existence and regularity of the smooth transonic steady solutions of Euler--Poisson equations representing for the hydrodynamic model of semiconductors. Different ...

We consider a population dynamics model with degenerate diffusion and time delay. We discover sharp-oscillatory waves with sharp edges and non-decaying oscillations arising from density-dependent dispersal and reproduction with age structure. Degenerate diffusion and bad effect of time delay prevent the use of existing approaches. Here, we develop a new delayed iteration framework to show the existence of these peculiar waves. In particular, the estimate of the admissible wave speed is highly nontrivial. We employ a new phase transform method coupled with the detailed analysis of phase energy. Furthermore, we give a complete characterization of the dynamical behaviors of various kinds of waves. Our results indicate that simple invasion rules can generate complex wave patterns and provide some interesting insights into the ecological dynamics.

As shown in [17] , [18] , for the hydrodynamic model of semiconductors represented by Euler-Poisson equations with sonic boundary and subsonic/supersonic doping profile, the structure of stationary solutions are very complicated. It may possess various solutions like subsonic/supersonic/transonic flows. In this paper, we consider a more challenging case where the doping profile is transonic, which is categorized into two types: subsonic-dominated and supersonic-dominated. In the subsonic-dominated case, we show that the system has a unique interior subsonic solution, at least one interior supersonic solution and infinitely many transonic solutions under the suitable assumptions. However, the difference with the case of subsonic doping profile is that the interior subsonic solution and interior supersonic solution may not exist in special cases when the relaxation time is small. In the supersonic-dominated case, the non-existence and existence of all types of solutions are also obtained. The approach adopted is the technical compactness analysis combining the Green's function method. Here, the results obtained perfectly develop the existing studies.

The aim of this paper is to give existence results for a class of coupled systems of fractional integrodifferential equations with Hilfer fractional derivative in Banach spaces. We first give some definitions, namely the Hilfer fractional derivative and the Hausdorff’s measure of noncompactness and the Sadovskii’s fixed point theorem.

For the classical reaction diffusion equation, the a priori speed of fronts is determined exactly in the pioneering paper (Benguria and Depassier 1996 Commun. Math. Phys. 175 221–227) by variational characterization method. In this paper, we study the age-structured population dynamics using a degenerate diffusion equation with time delay. We show the existence and uniqueness of sharp critical fronts, where the sharp critical front is C 1-smooth when the diffusion degeneracy is weaker with 1 < m < 2, and the sharp critical front is non-C 1-smooth (piecewise smooth) when the diffusion degeneracy is stronger with m ≥ 2, and the non-critical waves are C 2-smooth. We give a new variational approach for the critical wave speed and investigate how the time delay affects the propagation mechanism of fronts. It is shown that the time delay slows down the critical wave speed.

This paper deals with the Cauchy problem for the compressible Euler equations with time-dependent damping, where the time-vanishing damping in the form of μ ( 1 + t ) λ makes some fantastic variety of the dynamic system. For 0 λ 1 and μ > 0 , or λ = 1 but μ > 2 , the solutions are proved to exist globally in time, when the derivatives of the initial data are small, but the initial data themselves can be arbitrarily large. This is the so-called challenging case of global solutions with large initial data; while, when the initial Riemann invariants are monotonic and their derivatives with absolute value are large at least at one point, then the solutions are still bounded, but their derivatives will blow up at finite time, somewhat like the singularity formed by shock waves. For λ > 1 and μ > 0 , or λ = 1 but 0 μ ≤ 1 , the derivatives of solutions will blow up even for all initial data, including the interesting case of blow-up solutions with small initial data. Here the initial Riemann invariants are monotonic. In fact, such a blow-up phenomenon is determined by the mechanism of the dynamic system itself. In order to prove the global existence of solutions with large initial data, we introduce a new energy functional related to the Riemann invariants, which crucially enables us to build up the maximum principle for the corresponding Riemann invariants, and the uniform boundedness for the local solutions. Finally, numerical simulations in different cases are carried out, which further confirm our theoretical results.

In this paper, we are concerned with the system of Euler equations with time-dependent damping like $-\frac{\mu}{(1+t)^\lambda}u$ for physical parameters $\lambda\ge 0$ and $\mu>0$. It is well know...

We investigate the propagating profiles of a degenerate chemotaxis model describing the bacteria chemotaxis and consumption of oxygen by aerobic bacteria, in particular, the effect of the initial attractant distribution on bacterial clustering. We prove that the compact support of solutions may shrink if the signal concentration satisfies a special structure, and show the finite speed propagating property without assuming the special structure on attractant concentration, and obtain an explicit formula of the population spreading speed in terms of model parameters. The presented results suggest that bacterial cluster formation can be affected by chemotactic attractants and density-dependent dispersal.

In this paper, we study the one-dimensional Euler–Poisson equations of bipolar hydrodynamic model for semiconductor device with time-dependent damping effect − J ( 1 + t ) λ for − 1 λ 1 , where the damping effect is time-gradually-degenerate for λ > 0 , and time-gradually-enhancing for λ 0 . Such a damping effect makes the hydrodynamic system possess the nonlinear diffusion phenomena time-asymptotic-weakly or strongly. Based on technical observation, and by using the time-weighted energy method, where the weights are artfully chosen, we prove that the system admits a unique global smooth solution, which time-asymptotically converges to the corresponding diffusion wave, when the initial perturbation around the diffusion wave is small enough. The convergence rates are specified in the algebraic forms O ( t − 3 4 ( 1 + λ ) ) and O ( t − ( 1 − λ ) ) according to different values of λ in ( − 1 , 1 7 ) and ( 1 7 , 1 ) , respectively, where λ = 1 7 is the critical point, and the convergence rate at the critical point is O ( t − 6 7 ln ⁡ t ) . All these convergence rates obtained in different cases are optimal in the sense when the initial perturbations are L 2 -integrable. Particularly, when λ = 1 7 , the convergence rate is the fastest, namely, the asymptotic profile of the original system at the critical point is the best.

In this paper, we prove the existence, uniqueness and regularity of global solution to the bistable reaction–diffusion equation with degenerate diffusion and nonlocal time-delay. The adopted approach is the Holmgren’s approximation scheme combining the compactness analysis.

This paper is concerned with time-delayed reaction–diffusion equations with degenerate diffusion. When the term for birth rate is a nonlocal integral with a heat kernel, the family of minimum wave speeds corresponding to all the degenerate diffusion coefficients is proved to admit a uniform positive infimum. However, when the term for birth rate is local, there is no positive infimum of all the minimum wave speeds. This difference indicates that the nonlocal effect plays a role as Laplacian such that a positive lower bound independent of the degenerate diffusion exists for the minimum wave speeds. The approach adopted for the proof is the monotone technique with the viscosity vanishing method. The degeneracy of diffusion for the equation causes us essential difficulty in the proof. A number of numerical simulations are also carried out at the end of the paper, which further numerically confirm our theoretical results.

This paper deals with the existence and stability of traveling wave solutions for a degenerate reaction–diffusion equation with time delay. The degeneracy of spatial diffusion together with the effect of time delay causes us the essential difficulty for the existence of the traveling waves and their stabilities. In order to treat this case, we first show the existence of smooth- and sharp-type traveling wave solutions in the case of $$c\ge c^*$$ for the degenerate reaction–diffusion equation without delay, where $$c^*>0$$ is the critical wave speed of smooth traveling waves. Then, as a small perturbation, we obtain the existence of the smooth non-critical traveling waves for the degenerate diffusion equation with small time delay $$\tau >0$$ . Furthermore, we prove the global existence and uniqueness of $$C^{\alpha ,\beta }$$ -solution to the time-delayed degenerate reaction–diffusion equation via compactness analysis. Finally, by the weighted energy method, we prove that the smooth non-critical traveling wave is globally stable in the weighted $$L^1$$ -space. The exponential convergence rate is also derived.

In this paper, we derive a chemotaxis model with degenerate diffusion and density-dependent chemotactic sensitivity, and we provide a more realistic description of cell migration process for its early and late stages. Different from the existing studies focusing on the case of non-degenerate diffusion, this model with degenerate diffusion causes us some essential difficulty on the boundedness estimates and the propagation behavior of its compact support. In the presence of logistic damping, for the early stage before tumour cells spread to the whole domain, we first estimate the expanding speed of tumour region as O(t^β) for 0 < β < 1/2 . Then, for the late stage of cell migration, we further prove that the asymptotic profile of the original system is just its corresponding steady state. The global convergence of the original weak solution to the steady state with exponential rate O(e^(-ct)) for some c < 0 is also obtained.

The purpose of this paper is to study the multi-dimensional steady hydrodynamic model of semiconductors represented by Euler-Poisson equations with sonic boundary. We prove that, the steady Euler-Poisson equations with sonic boundary possess a unique subsonic solution and at least one supersonic solution in the radial form. The adopted approach for proof is the energy method combining the compactness analysis. For the n-D radial supersonic solutions, since it is more challenging to get the crucial energy estimates due to the effect by the multiple dimensions and the restriction by the sonic boundary, we propose a brand new two-steps iteration scheme to build up the key energy estimates. This is the first attempt to study the n-D steady-states with the sonic boundary, and the results obtained essentially improve and develop the previous studies in the one-dimensional case.

In this paper, we study the Cauchy problem for the hyperbolic p-system with time-gradually-degenerate damping term − 1 ( 1 + t ) λ u for 0 ⩽ λ 1 , and show that the damped p-system has a couple of global solutions uniquely, and such solutions tend time-asymptotically to the shifted nonlinear diffusion waves, which are the solutions of the corresponding nonlinear parabolic equation governed by the Darcy's law. We further derive the convergence rates when the initial perturbations are in L 2 . The approach adopted is the technical time-weighted energy method.

In this paper we study the hybrid quantum hydrodynamic model for nano-sized bipolar semiconductor devices in thermal equilibrium. By introducing a hybrid version of the Bhom potential, we derive a bipolar hybrid quantum hydrodynamic model, which is able to account for quantum effects in a localized region of the device for both electrons and holes. Coupled with Poisson equation for the electric potential, the steady-state system is regionally degenerate in its ellipticity, due to the quantum effect only in part of the device. This regional degeneracy of ellipticity makes the study more challenging. The main purpose of the paper is to investigate the existence and uniqueness of the weak solutions to this new type of equations. We first establish the uniform boundedness of the smooth solutions to the modified bipolar quantum hydrodynamic model by the variational method, then we use the compactness technique to prove the existence of weak solutions to the original hybrid system by taking hybrid limit. In particular, we account for two different kinds of hybrid behaviour. We perform the first hybrid limit when both electrons and holes behave quantum in a given region of the device, and the second one when only one carrier exhibits hybrid behaviour, whereas the other one is presented classically in the whole domain. The semi-classical limit results are also obtained. Finally, the theoretical results are tested numerically on a simple toy model.

The relaxation limit from bipolar semiconductor hydrodynamic (HD) model to drift–diffusion (DD) model is shown under the non-constant doping profile assumption for both stationary solutions and global-in-time solutions, which satisfy the general form of the Ohmic contact boundary condition. The initial layer phenomenon will be analyzed because the initial data is not necessarily in the momentum equilibrium. Due to the bipolar coupling structure, the analysis is hard and different from the previous literature on unipolar model or bipolar model with zero doping profile restriction. We first construct the non-constant uniform stationary solutions by the operator method for both HD and DD models in a unified procedure. Then we prove the global existence of DD model and uniform global existence of HD model by the elementary energy method but with some new developments. Based on the above existence results, we further calculate the convergence rates in relaxation limits.

This paper is concerned with the multi-dimensional compressible Euler equations with time-dependent over-damping of the form $-\frac{\mu}{(1+t)^\lambda}\rho\boldsymbol u$ in $\mathbb R^n$, where $n\ge2$, $\mu>0$, and $\lambda\in[-1,0)$. This continues our previous work dealing with the under-damping case for $\lambda\in[0,1)$. We show the optimal decay estimates of the solutions such that for $\lambda\in(-1,0)$ and $n\ge2$, $\|\rho-1\|_{L^2(\mathbb R^n)}\approx(1+t)^{-\frac{1+\lambda}{4}n}$ and $\|\boldsymbol u\|_{L^2(\mathbb R^n)}\approx (1+t)^{-\frac{1+\lambda}{4}n-\frac{1-\lambda}{2}}$, which indicates that a stronger damping gives rise to solutions decaying optimally slower. For the critical case of $\lambda=-1$, we prove the optimal logarithmical decay of the perturbation of density for the damped Euler equations such that $\|\rho-1\|_{L^2(\mathbb R^n)}\approx |\ln(e+t)|^{-\frac{n}{4}}$ and $\|\boldsymbol u\|_{L^2(\mathbb R^n)}\approx (1+t)^{-1}\cdot|\ln(e+t)|^{-\frac{n}{4}-\frac{1}{2}}$ for $n\ge7$. The over-damping effect reduces the decay rates of the solutions to be slow, which causes us some technical difficulty in obtaining the optimal decay rates by the Fourier analysis method and the Green function method. Here, we propose a new idea to overcome such a difficulty by artfully combining the Green function method and the time-weighted energy method.

Currently, the analytical analysis cannot describe the deformation and draping behavior during hemisphere preforming using plain woven fabrics, especially considering the fabric deformation outside the hemisphere and the effect of processing parameters. Thus, an analytical analysis was established to characterize the sliding and shearing behavior for a typical hemisphere preforming for the plain woven fabric. The experiments were conducted, and the root mean square values for the sliding and shearing between the analytical and experimental results are small, demonstrating the effectiveness of the analytical analysis. With the reduction of the corner radius of the die and the gap between the punch and die, the sliding ratio is larger than 1.0, indicating an increase in yarn sliding . In addition, the shearing deformation in the quasi tensile region is still generated and reaches a maximum value of 4.5° both in analytical prediction and in experimental measurements. Moreover, the analytical analysis is capable of characterizing the draping degree, which is shown to increase non-linearly with increasing punch depth. The reduction of the gap and corner radius enlarges the draped region, improving the draping degree. This work demonstrates that an analytical analysis can be used to gain an insight into the deformation and draping mechanisms of plain woven fabrics.

This short note is concerned with Nicholson’s blowflies equation with Dirichlet boundary. For the challenging non-monotone case, we show the exponential convergence rate by the energy method, where the key observation is that the energy integrations for the perturbations on the inner boundary will be disappeared automatically, even the solutions at the boundary are non-zero.

This paper is concerned with the Cauchy problem for the one-dimensional compressible Euler equations with critical time-dependent overdamping. In this critical case, the damped Euler system is proved to admit a unique global solution that converges time-asymptotically to the corresponding nonlinear diffusion wave. Particularly, the optimal convergence rate in logarithmic form is first derived by the technical time-weighted method.

This is the second part of our series of studies concerning the well-posedness/ill-posedness and regularity of stationary solutions to the hydrodynamic model of semiconductors represented by Euler--Poisson equations with sonic boundary condition. In this paper, we consider the case of a supersonic doping profile, and prove that the system does not hold any interior subsonic solution; furthermore, the system doesn't admit any interior supersonic solution and any transonic solution if such a supersonic doping profile is small enough or the relaxation time is small, but it has at least one interior supersonic solution and infinitely many transonic solutions if the supersonic doping profile is close to the sonic line and the relaxation time is large. The nonexistence of any type of solutions in the case of a small doping profile or small relaxation time indicates that the semiconductor effect for the system is remarkable and cannot be ignored.

In this paper, we study the asymptotic behaviors in time of solutions to the unipolar hydrodynamic model of semiconductors in the form of Euler–Poisson equations on the half line with the boundary effect, where the boundary conditions are proposed physically as the inflow/outflow/impermeable boundary or the insulating boundary. Different from the Cauchy problem, the boundary effect causes some essential difficulties in determining the asymptotic profiles for the solutions and occurs the boundary layers. First of all, by heuristically analyzing, we reasonably propose some additional boundary conditions at far field to the corresponding steady-state equations such that the steady-state systems are well-posed. Thus, we can determine the corresponding steady-states as the expected asymptotic profiles for the solutions of original IBVPs. Secondly, there are some L 2 -boundary-layers between the solutions of original IBVPs and their steady-states. After investigating the exact form of gaps, we technically construct some correction functions to delete these gaps. Finally, by using the energy estimates, we further prove that the original solutions of the inflow/outflow/impermeable problem (insulating problem) time-exponentially (time-exponentially/algebraically) converge to their asymptotic profiles. Finally, we carry out some numerical simulations, which show that, the graphs for the asymptotic profiles in different boundary cases are significantly distinct.

This paper is concerned with the time delayed Nicholson’s blowflies equation with degenerate diffusion. We prove the existence and uniqueness of the positive steady state solution under the Dirichlet boundary condition and we show the stability of the nontrivial steady state.

In this paper, we investigate the properties of traveling waves to a class of lattice differential equations for cellular neural networks with multiple delays. Following the previous study [38] on the existence of the traveling waves, here we focus on the uniqueness and the stability of these traveling waves. First of all, by establishing the a priori asymptotic behavior of traveling waves and applying Ikehara's theorem, we prove the uniqueness (up to translation) of traveling waves ϕ ( n − c t ) with c ≤ c ⁎ for the cellular neural networks with multiple delays, where c ⁎ 0 is the critical wave speed. Then, by the weighted energy method together with the squeezing technique, we further show the global stability of all non-critical traveling waves for this model, that is, for all monotone waves with the speed c c ⁎ , the original lattice solutions converge time-exponentially to the corresponding traveling waves, when the initial perturbations around the monotone traveling waves decay exponentially at far fields, but can be arbitrarily large in other locations.

This paper deals with a $p$-Kirchhoff type problem involving sign-changing weight functions. It is shown that under certain conditions, by means of variational methods, the existence of multiple nontrivial nonnegative solutions for the problem with the subcritical exponent are obtained. Moreover, in the case of critical exponent, we establish the existence of the solutions and prove that the elliptic equation possesses at least one nontrivial nonnegative solution.

This paper concerns a parabolic–hyperbolic system on the half space R+ with boundary effect. The system is derived from a singular chemotaxis model describing the initiation of tumor angiogenesis. We show that the solution of the system subject to appropriate boundary conditions converges to a traveling wave profile as time tends to infinity if the initial data is a small perturbation around the wave which is shifted far away from the boundary but its amplitude can be arbitrarily large.

This paper is concerned with the stability of non-monotone traveling waves to a nonlocal dispersion equation with time-delay, a time-delayed integro-differential equation. When the equation is crossing-monostable, the equation and the traveling waves both loss their monotonicity, and the traveling waves are oscillating as the time-delay is big. In this paper, we prove that all non-critical traveling waves (the wave speed is greater than the minimum speed), including those oscillatory waves, are time-exponentially stable, when the initial perturbations around the waves are small. The adopted approach is still the technical weighted-energy method but with a new development. Numerical simulations in different cases are also carried out, which further confirm our theoretical result. Finally, as a corollary of our stability result, we immediately obtain the uniqueness of the traveling waves for the non-monotone integro-differential equation, which was open so far as we know.

This paper is concerned with the stability of critical traveling waves for a kind of non-monotone time-delayed reaction–diffusion equations including Nicholson's blowflies equation which models the population dynamics of a single species with maturation delay. Such delayed reaction–diffusion equations possess monotone or oscillatory traveling waves. The latter occurs when the birth rate function is non-monotone and the time-delay is big. It has been shown that such traveling waves ϕ ( x + c t ) exist for all c ≥ c ⁎ and are exponentially stable for all wave speed c > c ⁎ [13] , where c ⁎ is called the critical wave speed. In this paper, we prove that the critical traveling waves ϕ ( x + c ⁎ t ) (monotone or oscillatory) are also time-asymptotically stable, when the initial perturbations are small in a certain weighted Sobolev norm. The adopted method is the technical weighted-energy method with some new flavors to handle the critical oscillatory waves. Finally, numerical simulations for various cases are carried out to support our theoretical results.

This paper is concerned with Nicholson's blowflies equation, a kind of time-delayed reaction-diffusion equation. It is known that when the ratio of birth rate coefficient and death rate coefficient satisfies $1 e$, the equation losses its monotonicity, and its traveling waves are oscillatory when the time-delay $r$ or the wave speed $c$ is large, which causes the study of stability of these nonmonotone traveling waves to be challenging. In this paper, we use the technical weighted energy method to prove that when $e c_*>0$ are exponentially stable, where $c_*>0$ is the minimum wave speed. Here, we allow the traveling wave to be either monotone or nonmonotone with any speed $c>c_*$ and any size of the time-delay $r>0$; however, when $\frac{p}{d}> e^2$ with a smal...