Search

We consider the semilinear wave equation with time varying delay and linear internal feedback u t t ( x , t ) − Δ u ( x , t ) + μ 1 u t ( x , t ) + μ 2 u t ( x , t − τ ( t ) ) + f ( x , u ) = g , in a regular bounded domain of R n with Dirichlet boundary conditions. Under suitable relations between μ 1 and μ 2 and suitable conditions on f and τ we derive first a boundedness result and we show that any global bounded solution of the above equation has a relative compact range in the natural energy space. More importantly, when the nonlinear term f is analytic in the second variable, we show the convergence to equilibrium as well as estimates for the rate of convergence for any global bounded solution. We also discuss existence of global solutions.

In lithium battery research, when we need to establish equivalent circuit based on impedance measurement, we can find several suitable circuit models. We usually use the chi-squared to determine the validity of equivalent models; when the chi-squared is less than 0.01, it indicates that those models have good fitting effect. In this case, we propose to verify the Kramers–Kronig transformation by extrapolating the data with equivalent circuit model and compare the residual errors of the Kramers–Kronig transform to determine the most suitable equivalent circuit model.

This paper develops a unified method to derive compactness properties and convergence to a steady state as well as decay estimates of global bounded solutions of the abstract semilinear second order integro-differential equation $$\begin{aligned} u_{tt}(t)+ Au(t) -\int ^\tau _0 k(s) Au(t-s)ds+ f(u)=g(t), \ \tau \in \{t, \infty \}, \end{aligned}$$ with finite ( $$\tau =t$$ ) or infinite ( $$\tau =\infty $$ ) memory. Here, A is a continuous linear operator, k is a positive nonincreasing function, f is a nonlinear function such that $$A+f$$ is the gradient of a potential satisfying the Łojasiewicz–Simon inequality, and the time-dependent right-hand side g decays to 0 at infinity. The general results are applied to particular semilinear evolution viscoelastic problems at the end of the paper.

By a small perturbation of the usual Lyapunov energy and by using the Łojasiewicz–Simon inequality, we show that the dissipation given by the memory term is strong enough to prove the convergence to equilibrium as well as estimates for the rate of convergence for any global bounded solution to the semilinear viscoelastic equation $$\begin{aligned} |u_t |^\rho u_{tt} -\Delta u_{tt}-\Delta u +\int ^\tau _0 k(s) \Delta u(t-s)\mathrm{d}s+ f(x,u)=g, \ \tau \in \{t, \infty \}, \end{aligned}$$ in a bounded regular domain of \({\mathbb {R}}^n\) with Dirichlet boundary conditions. Here, the kernel function \(k > 0\) is assumed to decay exponentially at infinity, the nonlinearity f is analytic in the second variable, and the forcing term g is supposed to decay polynomially or exponentially at infinity. The present work extends the previous result where the given equation was studied in the presence of an additional strong linear damping \(-\,\Delta u_t\).

In this paper we consider the nonautonomous semilinear viscoelastic equation u t t − Δ u + ∫ 0 τ k ( s ) Δ u ( t − s ) d s + f ( x , u ) = g , τ ∈ { t , ∞ } , in R + × Ω , with Dirichlet boundary conditions and finite ( τ = t ) or infinite ( τ = ∞ ) memory. Here Ω is a bounded domain in R n with smooth boundary and the nonlinearity f : Ω × R + → R is analytic in the second variable, uniformly with respect to the first one. For this equation, we derive an appropriate Lyapunov function and we use the Łojasiewicz–Simon inequality to show that the dissipation given by the memory term is strong enough to prove the convergence to a steady state for any global bounded solution. In addition, we discuss the rate of convergence to equilibrium which is polynomial or exponential, depending on the Łojasiewicz exponent and the decay of the time-dependent right-hand side g.

We study the long-time behavior as time goes to infinity of global bounded solutions to the following nonautonomous semilinear viscoelastic equation: $$\begin{aligned} |u_t |^\rho u_{tt} -\Delta u_{tt}-\Delta u_{t}-\Delta u +\int ^\tau _0 k(s) \Delta u(t-s)ds+ f(x,u)=g, \ \tau \in \{t, \infty \}, \end{aligned}$$|ut|?utt-Δutt-Δut-Δu+?0?k(s)Δu(t-s)ds+f(x,u)=g,??{t,?},in $${\mathbb {R}}^+\times \Omega $$R+×Ω, with Dirichlet boundary conditions, where $$\Omega $$Ω is a bounded domain in $${\mathbb {R}}^n$$Rn and the nonlinearity f is analytic. Based on an appropriate (perturbed) new Lyapunov function and the ?ojasiewicz---Simon inequality we prove that any global bounded solution converges to a steady state. We discuss also the rate of convergence which is polynomial or exponential, depending on the ?ojasiewicz exponent and the decay of the term g.

Clip domain serine protease homologs are widely distributed in insect genomes and play important roles in regulating insect immune responses, yet their exact functions remain poorly understood. Here, we show that CLIPA2, a clip domain serine protease homolog of Anopheles gambiae, regulates the consumption of the mosquito complement-like protein TEP1 during systemic bacterial infections. We provide evidence that CLIPA2 localizes to microbial surfaces in a TEP1-dependent manner whereby it negatively regulates the activity of a putative TEP1 convertase, which converts the full-length TEP1-F form into active TEP1cut. CLIPA2 silencing triggers an exacerbated TEP1-mediated response that significantly enhances mosquito resistance to infections with a broad class of microorganisms including Plasmodium berghei, Escherichia coli and the entomopathogenic fungus Beauveria bassiana. We also provide further evidence for the existence of a functional link between TEP1 and activation of hemolymph prophenoloxidase during systemic infections. Interestingly, the enhanced TEP1-mediated immune response in CLIPA2 knockdown mosquitoes correlated with a significant reduction in fecundity, corroborating the existence of a trade-off between immunity and reproduction. In sum, CLIPA2 is an integral regulatory component of the mosquito complement-like pathway which functions to prevent an overwhelming response by the host in response to systemic infections.

Our aim in this article is to study a nonautonomous semilinear wave equation with nonlinear damping and dynamical boundary condition. First we prove the existence and uniqueness of global bounded solutions having relatively compact range in the natural energy space. Then, by deriving an appropriate Lyapunov energy, we show that if the exponent in the Łojasiewicz-Simon inequality is large enough (depending on the damping), then weak solutions converge to equilibrium.

In this article we present a unified approach to study the asymptotic behavior and the decay rate to a steady state of bounded weak solutions of nonlinear, gradient-like evolution equations of mixed first and second order. The proof of convergence is based on the Lojasiewicz–Simon inequality, the construction of an appropriate Lyapunov functional, and some differential inequalities. Applications are given to nonautonomous semilinear wave and heat equations with dissipative, dynamical boundary conditions, a nonlinear hyperbolic–parabolic partial differential equation, a damped wave equation and some coupled system.

The successful development of Plasmodium in Anopheles mosquitoes is governed by complex molecular and cellular interactions that we are just beginning to understand. Anopheles immune system has received particular attention as genetic evidence points clearly to its critical role in eliminating the majority of parasites invading the midgut epithelium. Several factors regulating Plasmodium development have been identified and tentatively assigned to the individual steps leading to mosquito immune reactions; non-self-recognition, signal modulation, signal transduction and effector mechanisms. Detailed knowledge of these steps and their underlying molecular mechanisms may offer novel perspectives to abort Plasmodium development in the vector. Here, we summarize our current knowledge of mosquito innate immunity highlighting both, recent advances and areas where additional research is required.

C-type lectins (CTLs) are a family of proteins that share a common structural motif, the carbohydrate recognition domain, and may act as receptors in pathogen recognition. Indeed, some vertebrate CTLs, particularly the collectins, are unequivocally implicated in the innate immune response to certain microbes. Although studies in insects and other invertebrates have described CTL activation of effector immune responses in vitro, the contribution of these CTLs to immune defenses in vivo is still poorly understood. Here we report that two CTLs, CTL4 and CTLMA2, which were shown previously to inhibit Plasmodium berghei ookinete melanization in the malaria vector Anopheles gambiae, are transcriptionally induced by bacterial challenge. Using in vivo reverse genetic analysis, we show that both CTLs are required for the clearance of Escherichia coli, but not Staphylococcus aureus, from adult female mosquitoes. Silencing either CTL dramatically reduces mosquito survival to Gram-negative but not to Gram-positive bacterial infections, suggesting a role in defense against Gram-negative bacteria. Furthermore, molecular characterization reveals that both CTLs are secreted into the mosquito hemolymph mainly in the form of a disulfide-linked heterodimer. This association explains the similar roles of these CTLs in bacterial defense as well as in the melanization response to P. berghei ookinetes. Apparently, CTL4 and CTLMA2 serve pleiotropic functions in the innate immune response of A. gambiae.

We consider a nonautonomous, semilinear, hyperbolic-parabolic equation subject to a dynamical boundary condition of memory type. First we prove the existence and uniqueness of global bounded solutions having relatively compact range in the natural energy space. Under the assumption that the nonlinear term $f$ is real analytic, we then derive an appropriate Lyapunov energy and we use the {\L}ojasiewicz-Simon inequality to show the convergence of global weak solutions to single steady states as time tends to infinity. Finally, we provide an estimate for the convergence rate.

The complement C3-like protein TEP1 of the mosquito Anopheles gambiae is required for defense against malaria parasites and bacteria. Two forms of TEP1 are present in the mosquito hemolymph, the full-length TEP1-F and the proteolytically processed TEP1cut that is part of a complex including the leucine-rich repeat proteins LRIM1 and APL1C. Here we show that the non-catalytic serine protease SPCLIP1 is a key regulator of the complement-like pathway. SPCLIP1 is required for accumulation of TEP1 on microbial surfaces, a reaction that leads to lysis of malaria parasites or triggers activation of a cascade culminating with melanization of malaria parasites and bacteria. We also demonstrate that the two forms of TEP1 have distinct roles in the complement-like pathway and provide the first evidence for a complement convertase-like cascade in insects analogous to that in vertebrates. Our findings establish that core principles of complement activation are conserved throughout the evolution of animals., Author Summary Mosquitoes are vectors of numerous human diseases including malaria. Disease transmission requires that microbes overcome the robust mosquito immune system. In the African malaria mosquito, the TEP1 protein that is homologous to mammalian complement factor C3 is shown to play a central role in mosquito immunity to malaria parasites and bacteria. In this study, we report that another mosquito protein belonging to a class of non-catalytic enzymes that are specific to arthropods is a core component of the mosquito complement-like immune pathway. We found that this new protein, named SPCLIP1, regulates the accumulation of TEP1 on malaria parasites and bacteria, and show that this can lead to distinct defense reactions including lysis and melanization of the pathogen. This work is valuable because it reveals novel insight into the regulation of mosquito complement on microbial surfaces such as those of the malaria parasites. Unraveling the molecular mechanisms regulating these defense responses may ultimately lead to the design of novel disease blocking strategies in the vector.