Search

We investigate models for nonlinear ultrasound propagation in soft biological tissue based on the one that serves as the core for the software package k‐Wave. The systems are solved for the acoustic particle velocity, mass density, and acoustic pressure and involve a fractional absorption operator. We first consider a system that incorporates additional viscosity in the equation for momentum conservation. By constructing a Galerkin approximation procedure, we prove the local existence of its solutions. In view of inverse problems arising from imaging tasks, the theory allows for the variable background mass density, speed of sound, and the nonlinearity parameter in the systems. Second, under stronger conditions on the data, we take the vanishing viscosity limit of the problem, thereby rigorously establishing the existence of solutions for the limiting system as well. [ABSTRACT FROM AUTHOR]

We consider a system of two nonlocal and nonlinear partial differential equations that describe some aspects of yeast cell-cell communication. We study local and global existence and uniqueness of solutions. We consider mild solutions and we perform bilinear and trilinear fixed point arguments in suitable functional spaces. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the following Keller–Segel system with flux limitation u t = ∇ ⋅ u ∇ u u 2 + | ∇ u | 2 − χ ∇ ⋅ (u f (| ∇ v | 2) ∇ v) , x ∈ Ω ,   t > 0 , 0 = Δ v − μ + u , x ∈ Ω ,   t > 0 , (⋆) under no-flux boundary conditions in a ball Ω = B R (0) ⊂ ℝ n (n ≥ 1), where μ : = 1 | Ω | ∫ Ω u 0 (x) d x is positive and f (ξ) = (1 + ξ) − α , for ξ ≥ 0 with α > 0. It is proved that the problem (⋆) possesses a unique classical solution that can be extended in time up to a maximal T max ∈ (0 , ∞ ]. Moreover, it is shown that the above solution is global and bounded when either χ < (1 + m 2) α m if n = 1 and α ≤ 1 2 , or χ < 2 α − 1 (2 α 2 α − 1) α if n ≥ 1 and α > 1 2 . We point out that when α = 1 2 , our result is consistent with that of [N. Bellomo and M. Winkler, Comm. Partial Differential Equations 42 (2017) 436–473]. [ABSTRACT FROM AUTHOR]

We are interested in the behavior of solutions to the damped inhomogeneous nonlinear Schrödinger equation $ i\partial_tu+\Delta u+\mu|x|^{-b}|u|^{\alpha}u+iau = 0 $, $ \mu \in{\mathbb C} $, $ b>0 $, $ {a}>0 $, $ \alpha>0 $. We establish lower and upper bound estimates of the life-span. We obtain explicit values $ a^*, \; a_* $ such that if $ aa^*, $ global existence holds. Also, we prove scattering results with precised decay rates for large damping. Some of the results are new even for $ b = 0. $ [ABSTRACT FROM AUTHOR]

For this study, we investigate the existence and uniqueness of local solutions and derive a blow-up solution for a quasi-linear bi-hyperbolic equation under dynamic boundary conditions. We utilize the contraction mapping concept to demonstrate the solution's local well-posedness and employ a concavity approach to establish the blow-up result.

In this paper, the motion of strictly convex closed plane curves under dissipative hyperbolic mean curvature flow is studied. The hyperbolic Monge–Amp è$$ \overset{\grave }{e} $$re equation is derived by using the support function. The short‐time existence of the flow is proved, and some evolution equations are derived. Furthermore, according to different initial velocities, we discuss the expansion and contraction of the dissipative hyperbolic curvature flow; that is, if the initial velocity v˜0<−α2ω0(θ)$$ {\tilde{v}}_0<-\frac{\alpha }{2}{\omega}_0\left(\theta \right) $$, the flow will converge to the limit curve at a finite time; if the initial velocity v˜0=−α2ω0(θ)$$ {\tilde{v}}_0&#x0003D;-\frac{\alpha }{2}{\omega}_0\left(\theta \right) $$, the flow will converge to a point t→∞$$ t\to \infty $$; if the initial velocity −−α2ω0(θ)0$$ {\tilde{v}}_0\left(\theta \right)>0 $$, the flow will expand to a limit curve as t→∞$$ t\to \infty $$. [ABSTRACT FROM AUTHOR]

In this paper we establish new and optimal estimates for the existence time of the maximal solutions to the nonlinear parabolic system ∂ t u = Δ u + | v | p - 1 v , ∂ t v = Δ v + | u | q - 1 u , q ≥ p ≥ 1 , q > 1 with initial values in Lebesgue or weighted Lebesgue spaces. The lower-bound estimates hold without any restriction on the sign or the size of the components of the initial data. To prove the upper-bound estimates, necessary conditions for the existence of nonnegative solutions are established. These necessary conditions allow us to give new sufficient conditions for finite time blow-up with initial values having critical decay at infinity. [ABSTRACT FROM AUTHOR]

In this paper, we study the Cauchy problem of the two-species incompressible viscoelastic fluid of Oldroyd-B system, which involving a reaction effect between two species of polymers. We prove the local existence with initial data in Hk(k ≥ 2) in a classical solution framework, and then provide a blow-up criteria. We concentrate on the a priori estimate, by using the energy method. In particular, the variant system in a general formulation is also studied, and the corresponding local well-posedness is established. [ABSTRACT FROM AUTHOR]

In this paper, a class of viscoelastic evolution equation on manifolds with conical singularities is concerned. Local existence and uniqueness of the weak solution are established by making use of the Faedo-Galerkin method and the contraction mapping principle. Moreover, by constructing an auxiliary functional and exploiting Levine's concavity method, the finite time blow-up criteria are obtained for arbitrary initial energy. In particular, the upper and lower bound of the blow-up time are also derived. [ABSTRACT FROM AUTHOR]

The local existence of solutions to nonhomogeneous Navier–Stokes equations in cylindrical domains with arbitrary large flux is demonstrated. The existence is proved by the method of successive approximations. To show the existence with the lowest possible regularity the special Besov spaces called the Sobolev–Slobodetskii spaces are used. The inflow and outflow are prescribed on the parts of the boundary which are perpendicular to the x3$$ {x}_3 $$‐axis. Since the inflow and outflow are positive the crucial point of this paper is to verify that x3$$ {x}_3 $$‐coordinate of velocity is also positive. Finally, we conclude the local existence such that the velocity belongs to Wσ2+s,1+s/2(Ωt)$$ {W}_{\sigma}&#x0005E;{2&#x0002B;s,1&#x0002B;s/2}\left({\Omega}&#x0005E;t\right) $$, the gradient of pressure to Wσs,s/2(Ωt)$$ {W}_{\sigma}&#x0005E;{s,s/2}\left({\Omega}&#x0005E;t\right) $$ and the density to Wr,∞1,1(Ωt)$$ {W}_{r,\infty}&#x0005E;{1,1}\left({\Omega}&#x0005E;t\right) $$, where s∈(0,1),σ>3/s,r>5/s,r>σ$$ s\in \left(0,1\right),\sigma >3/s,r>5/s,r>\sigma $$. [ABSTRACT FROM AUTHOR]

For this study, we investigate the existence and uniqueness of local solutions and derive a blow-up solution for a quasi-linear bi-hyperbolic equation under dynamic boundary conditions. We utilize the contraction mapping concept to demonstrate the solution's local well-posedness and employ a concavity approach to establish the blow-up result. [ABSTRACT FROM AUTHOR]

We study a porous medium equation with dissipative and hyperdiffusive effects and no sign restrictions on the diffusion coefficient. [ABSTRACT FROM AUTHOR]

This paper devotes to consider the local existence of the strong solutions to the generalized MHD system with fractional dissipative terms 2au for the velocity field and 2ab for the magnetic field, respectively. We construct the approximate solutions by the Fourier truncation method, and use energy method to obtain the local existence of strong solutions in Hs(Rn) (s > max n 2 + 1 - 2a, 0) for any a = 0. [ABSTRACT FROM AUTHOR]

In the context of the initial data and an amplitude parameter ε , we establish a local existence result for a highly nonlinear shallow water equation on the real line. This result holds in the space H k as long as k > 5 / 2 . Additionally, we illustrate that the threshold time for the occurrence of wave breaking in the surging type is on the order of ε - 1 , while plunging breakers do not manifest. Lastly, in accordance with ODE theory, it is demonstrated that there are no exact solitary wave solutions in the form of sech and s e c h 2 . [ABSTRACT FROM AUTHOR]

This paper concerns a viscoelastic Kirchhoff-type equation with the dispersive term, internal damping, and logarithmic nonlinearity. We prove the local existence of a weak solution via a modified lemma of contraction of the Banach fixed-point theorem. Although the uniqueness of a weak solution is still an open problem, we proved uniqueness locally for specifically suitable exponents. Furthermore, we established a result for local existence without guaranteeing uniqueness, stating a contraction lemma. [ABSTRACT FROM AUTHOR]

In this article, our focus lies in investigating the existence of global solutions and the occurrence of infinite-time blowup for a nonlinear Klein-Gordon equation characterized by logarithmic nonlinearity, specifically in the form wlog |w|k. Notably, our inquiry involves handling logarithmic functions within the reaction terms and accommodating the functions w0 and w1 in the boundary terms. Consequently, a pivotal task is to establish blowup conditions that intricately hinge on the characteristics of the domains and the boundary conditions. It is of significant note that our exploration incorporates domain and boundary information into the formulation of blowup conditions. By employing a combination of the potential well technique and energy estimation methodology, we delve into scenarios of low initial energy and critical initial energy. In doing so, we derive a set of sufficient conditions that encompass both the global existence and the potential explosive behaviour of solutions pertaining to this variant of the Klein-Gordon equation. This work contributes to enhancing our understanding of the intricate interplay between logarithmic nonlinearity, domain characteristics, and boundary conditions in shaping the behaviour of solutions in the realm of nonlinear wave equations. [ABSTRACT FROM AUTHOR]

This article mainly deals with the blow-up properties of nonnegative solutions for a reaction–diffusion system coupled with norm-type sources under positive boundary value conditions. The local existence of a nonnegative solution and the comparison principle are given. The criteria for the global existence or finite time blow-up of the solutions are obtained by constructing new functions and utilizing the super- and -sub-solution method. The results reveal a correlation between the blow-up profiles of the solutions and the size of the domain, as well as the positive boundary value. [ABSTRACT FROM AUTHOR]

This paper concerns a viscoelastic Kirchhoff-type equation with the dispersive term, internal damping, and logarithmic nonlinearity. We prove the local existence of a weak solution via a modified lemma of contraction of the Banach fixed-point theorem. Although the uniqueness of a weak solution is still an open problem, we proved uniqueness locally for specifically suitable exponents. Furthermore, we established a result for local existence without guaranteeing uniqueness, stating a contraction lemma.

In this article, we investigate the initial-boundary value problem for a class of finitely degenerate semilinear parabolic equations with singular potential term. By applying the Galerkin method and Banach fixed theorem we establish the local existence and uniqueness of the weak solution. On the other hand, by constructing a family of potential wells, we prove the global existence, the decay estimate and the finite time blow-up of solutions with subcritical or critical initial energy.

The Cauchy problem in Rn{{\mathbb{R}}}^{n}, n≥2n\ge 2, for ut=Δu−∇⋅(uS⋅∇v),0=Δv+u,(⋆)\begin{array}{r}\left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{l}{u}_{t}=\Delta u-\nabla \cdot \left(uS\cdot \nabla v),\\ 0=\Delta v+u,\end{array}\right.\hspace{2.0em}\hspace{2.0em}\hspace{2.0em}\left(\star )\end{array} is considered for general matrices S∈Rn×nS\in {{\mathbb{R}}}^{n\times n}. A theory of local-in-time classical existence and extensibility is developed in a framework that differs from those considered in large parts of the literature by involving bounded classical solutions. Specifically, it is shown that for all non-negative initial data belonging to BUC(Rn)∩Lp(Rn){\rm{BUC}}\left({{\mathbb{R}}}^{n})\cap {L}^{p}\left({{\mathbb{R}}}^{n}) with some p∈[1,n)p\in \left[1,n), there exist Tmax∈(0,∞]{T}_{\max }\in \left(0,\infty ] and a uniquely determined u∈C0([0,Tmax);BUC(Rn))∩C0([0,Tmax);Lp(Rn))∩C∞(Rn×(0,Tmax))u\in {C}^{0}\left(\left[0,{T}_{\max });\hspace{0.33em}{\rm{BUC}}\left({{\mathbb{R}}}^{n}))\cap {C}^{0}\left(\left[0,{T}_{\max });\hspace{0.33em}{L}^{p}\left({{\mathbb{R}}}^{n}))\cap {C}^{\infty }\left({{\mathbb{R}}}^{n}\times \left(0,{T}_{\max })) such that with v≔Γ⋆uv:= \Gamma \star u, and with Γ\Gamma denoting the Newtonian kernel on Rn{{\mathbb{R}}}^{n}, the pair (u,v)\left(u,v) forms a classical solution of (⋆\star ) in Rn×(0,Tmax){{\mathbb{R}}}^{n}\times \left(0,{T}_{\max }), which has the property that ifTmax

In this article, our focus lies in investigating the existence of global solutions and the occurrence of infinite-time blowup for a nonlinear Klein–Gordon equation characterized by logarithmic nonlinearity, specifically in the form $ \mathbf {w}\log \vert \mathbf {w}\vert ^k $ . Notably, our inquiry involves handling logarithmic functions within the reaction terms and accommodating the functions $ \mathbf {w}_0 $ and $ \mathbf {w}_1 $ in the boundary terms. Consequently, a pivotal task is to establish blowup conditions that intricately hinge on the characteristics of the domains and the boundary conditions. It is of significant note that our exploration incorporates domain and boundary information into the formulation of blowup conditions. By employing a combination of the potential well technique and energy estimation methodology, we delve into scenarios of low initial energy and critical initial energy. In doing so, we derive a set of sufficient conditions that encompass both the global existence and the potential explosive behaviour of solutions pertaining to this variant of the Klein–Gordon equation. This work contributes to enhancing our understanding of the intricate interplay between logarithmic nonlinearity, domain characteristics, and boundary conditions in shaping the behaviour of solutions in the realm of nonlinear wave equations.

We consider the spatially inhomogeneous Landau equation with hard potentials (i.e., with γ Є [0, 1]) on the whole space R³. We prove the existence and uniquenss of solutions for a small time, assuming that the initial data are in a weighted tenth-order Sobolev space and have exponential decay in the velocity variable. In constrast to the soft potential case, local existence for the hard potentials case has been missing from the literature. This is because the moment loss issue is the most severe for these potentials. To get over this issue, our proof relies on a weighted hierarchy of norms that depends on the number of spatial and velocity derivatives in an asymmetric way. This hierarchy lets us take care of the terms that are affected by the moment loss issue the most. These terms do not give in to methods applied to study the existence of solutions to the Landau equation with soft potentials and are a major reason why the local existence problem was not known for the case of hard potentials. [ABSTRACT FROM AUTHOR]

In this work, we are interested in a logarithmic nonlinear viscoelastic wave equation with a time-varying delay term in a bounded domain. Using the Faedo-Galerkin approximation, we get the local existence of the solution. Then, under suitable conditions, we prove that solutions blow up in finite time. [ABSTRACT FROM AUTHOR]

Wave equations describing a wide variety of wave phenomena are commonly seen in mathematical physics. The inclusion of a noise term in a deterministic wave equation allows neglected degrees of freedom or fluctuations of external fields describing the environment to be considered in the equation. Moreover, adding a noise term to the deterministic equation reveals remarkable new features in the qualitative behavior of the solution. For example, noise can lead to singularities in some equations and prevent singularities in others. Taking into account the effects of the fluctuations along with a space-time white noise, we consider a relativistic wave equation with weak and strong damping terms and investigate the effect of multiplicative noise on the behavior of solutions. The existence of local and global solutions is provided, and some qualitative properties of solutions, such as continuous dependence of solutions on initial data, and blow up of solutions, are given. Moreover, an upper bound is provided for the blow up time.

In this paper, we study the following initial-boundary value problem of a three species predator-prey system with prey-taxis which describes the indirect prey interactions through a shared predator, i.e., $ \begin{align*} \begin{cases} u_t = d\Delta u+u(1-u)- \frac{a_1uw}{1+a_2u+a_3v}, & \; \mbox{in}\ \ \Omega, t>0, \\ v_t = \eta d\Delta v+rv(1-v)- \frac{a_4vw}{1+a_2u+a_3v}, & \; \mbox{in}\ \ \Omega, t>0, \\ w_t = \nabla\cdot(\nabla w-\chi_1 w\nabla u-\chi_2 w\nabla v) -\mu w+ \frac{a_5uw}{1+a_2u+a_3v}+\frac{a_6vw}{1+a_2u+a_3v}, & \mbox{in}\ \ \Omega, t>0, \ \ \label{II} \end{cases} \end{align*} $ under homogeneous Neumann boundary conditions in a bounded domain $ \Omega\subset \mathbb{R}^n (n \geqslant 1) $ with smooth boundary, where the parameters $ d, \eta, r, \mu, \chi_1, \chi_2, a_i > 0, i = 1, \ldots, 6. $ We first establish the global existence and uniform-in-time boundedness of solutions in any dimensional bounded domain under certain conditions. Moreover, we prove the global stability of the prey-only state and coexistence steady state by using Lyapunov functionals and LaSalle's invariance principle.

This note is to fix some small gaps in the proofs of some of the results in our paper [R.B. Salako and W. Shen, Journal of Differential Equations 262 (2017) 5635–5690]. [ABSTRACT FROM AUTHOR]

Wave equations describing a wide variety of wave phenomena are commonly seen in mathematical physics. The inclusion of a noise term in a deterministic wave equation allows neglected degrees of freedom or fluctuations of external fields describing the environment to be considered in the equation. Moreover, adding a noise term to the deterministic equation reveals remarkable new features in the qualitative behavior of the solution. For example, noise can lead to singularities in some equations and prevent singularities in others. Taking into account the effects of the fluctuations along with a space-time white noise, we consider a relativistic wave equation with weak and strong damping terms and investigate the effect of multiplicative noise on the behavior of solutions. The existence of local and global solutions is provided, and some qualitative properties of solutions, such as continuous dependence of solutions on initial data, and blow up of solutions, are given. Moreover, an upper bound is provided for the blow up time. [ABSTRACT FROM AUTHOR]

A nonlinear Cauchy problem of a wave equation with variable exponents is considered. The damping term is subjected to a coefficient of both variables space and time. We prove the local existence of solutions using the Galerkin approach while the blow-up result is obtained via the perturbed energy method. We finally end our paper with some numerical illustrations. Problems that are frequently used in the literature are special cases. [ABSTRACT FROM AUTHOR]

The paper is devoted to the study of subdiffusion equations with Mittag-Leffler nonlinearity. The comparison principle is proved in a bounded domain. The results on the local existence, global existence, and blow-up of solutions to the initial-boundary value problem are obtained. In addition, the results of local existence and blow-up of solutions to the initial problem on the whole Euclidean space are proven. [ABSTRACT FROM AUTHOR]

This paper is concerned with blow up of weak solutions of the following nonlinear viscoelastic problem with internal damping and logarithmic source term |ut|ρutt + M(∥u∥2)(-∆u) - ∆utt + Z0t g(t - s)∆u(s)ds + ut = u|u|p R-2 ln |u|k R with Dirichlet boundary initial conditions in a bounded domain Ω ⊂ Rn. In the physical point of view, this is a type of problems that usually arises in viscoelasticity. It has been considered with power source term first by Dafermos [3], in 1970, where the general decay was discussed. We establish conditions of p, ρ and the relaxation function g, for that the solutions blow up in finite time for positive and nonpositive initial energy. We extend the result in [15] where is considered M = 1 and external force type |u|p-2u in it. Further we state and sketch the proof of a result of local existence of weak solution that is used in the proof of the theorem on blow up. The idea underlying the proof of local existence of solution is based on Faedo-Galerkin method combined with the Banach fixed point method.

Abstract In this article, we consider a viscoelastic plate equation with past history, nonlinear damping, and logarithmic nonlinearity. We prove explicit and general decay rate results of the solution to the viscoelastic plate equation with past history. Convex properties, logarithmic inequalities, and generalized Young’s inequality are mainly used to prove the decay estimate.

The Cauchy problem for three-dimensional (3D) isentropic compressible radiation hydrodynamic equations is considered. When both shear and bulk viscosity coefficients depend on the mass density ρ\rho in a power law ρδ{\rho }^{\delta } (with 0

We consider the model for one‐dimensional micropolar, viscous, polytropic, and thermally conductive real gas flow with homogeneous boundary conditions, using the generalized equation of state for pressure. The generalization is shown by the fact that the pressure depends on the mass density as a power function. The governing system of partial differential equations is given in the Lagrangian description. Using the Faedo–Galerkin method and a priori estimates, we prove that the generalized solution exists locally in time. [ABSTRACT FROM AUTHOR]

The aim of this paper is to utilize special skills and potential well method to deal with questions of local and global well-posedness to a nonlinear diffusion equation driven by the p-Laplace operator, which reveals a subtle relationship between singular potential and nonlinear operator, and further generalizes some previous early results about local existence of the weak solution from a class of evolution equations involved the linear dissipative term and singular potential. Specifically, the results of stability and blow-up of solutions are established for the first time to this initial boundary value problem with singular potential. [ABSTRACT FROM AUTHOR]

This article mainly deals with the blow-up properties of nonnegative solutions for a reaction–diffusion system coupled with norm-type sources under positive boundary value conditions. The local existence of a nonnegative solution and the comparison principle are given. The criteria for the global existence or finite time blow-up of the solutions are obtained by constructing new functions and utilizing the super- and -sub-solution method. The results reveal a correlation between the blow-up profiles of the solutions and the size of the domain, as well as the positive boundary value.

We study a system of nonlinear wave equations of the Kirchhoff-Carrier type containing a variant of the Balakrishnan-Taylor damping in nonlinear terms. By the linearization method together with the Faedo-Galerkin method, we prove the local existence and uniqueness of a weak solution. On the other hand, by constructing a suitable Lyapunov functional, a sufficient condition is also established to obtain the exponential decay of weak solutions.

The Cauchy problem for three-dimensional (3D) isentropic compressible radiation hydrodynamic equations is considered. When both shear and bulk viscosity coefficients depend on the mass density ρ in a power law ρ δ (with 0 < δ < 1), based on some elaborate analysis of this system's intrinsic singular structures, we establish the local-in-time well-posedness of regular solution with arbitrarily large initial data and far field vacuum in some inhomogeneous Sobolev spaces by introducing some new variables and initial compatibility conditions. Note that due to the appearance of the vacuum, the momentum equations are degenerate both in the time evolution and viscous stress tensor, which, along with the strong coupling between the fluid and the radiation field, make the study on corresponding well-posedness challenging. For proving the existence, we first introduce an enlarged reformulated structure by considering some new variables, which can transfer the degeneracies of the radiation hydrodynamic equations to the possible singularities of some special source terms, and then carry out some singularly weighted energy estimates carefully designed for this reformulated system. [ABSTRACT FROM AUTHOR]

In this paper, the Cauchy problem for an integro-differential diffusion equation and a system with nonlocal nonlinear sources are considered. The results on the existence of local integral solutions and the nonexistence of global weak solutions to the nonlinear integro-differential diffusion equation and system are presented. [ABSTRACT FROM AUTHOR]

A boundary value problem is formulated for a stationary model of mass transfer, which generalizes the Boussinesq approximation in the case when the coefficients in the model equations can depend on the concentration of a substance or on spatial variables. The global existence of a weak solution of this boundary value problem is proved. Some fundamental properties of its solutions are established. In particular, the validity of the maximum principle for the substance's concentration has been proved. Sufficient conditions on the input data of the boundary value problem under consideration, which ensure the local existence of the strong solution from the space H 2 , and conditions that ensure the conditional uniqueness of the weak solution with additional property of smoothness for the substance's concentration are established. [ABSTRACT FROM AUTHOR]

The paper investigates a class of a semilinear wave equation with time-dependent damping term (-1/(1+t)β Δut) and a nonlinearity |u|p. We will show the inuence of the parameter β in the blow-up results under some hy-pothesis on the initial data and the exponent p by using the test function method. We also study the local existence in time of mild solution in the energy space H¹(Rn) × L²(Rn). [ABSTRACT FROM AUTHOR]

In this article, we consider a viscoelastic plate equation with past history, nonlinear damping, and logarithmic nonlinearity. We prove explicit and general decay rate results of the solution to the viscoelastic plate equation with past history. Convex properties, logarithmic inequalities, and generalized Young's inequality are mainly used to prove the decay estimate. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Mathematics, Mechanics & Computer Science is the property of Al-Farabi Kazakh National University and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We study the homogeneous Dirichlet problem for the degenerate parabolic equation of the Kirchhoff type ut−a(‖∇u‖22)Δu=b(‖u‖22)|u|q(x,t)−2uinQT=Ω×(0,T),where T > 0, Ω⊂ℝn, n ≥ 2, is a smooth bounded domain. The exponent q(x, t) is a measurable function in QT with values in an interval [q−, q+] ⊂ (1, ∞). The coefficients a(·), b(·) are continuous functions defined on ℝ+. It is assumed that a(s) → 0 or a(s) → ∞ as s → 0+; therefore, the equation degenerates or becomes singular as ‖ ∇ u(t)‖2 → 0. We prove the local in time solvability of the problem and derive sufficient conditions of the finite time blow‐up of the nonnegative solutions. The upper and lower estimates on the blow‐up moment are found. [ABSTRACT FROM AUTHOR]

In this paper, we consider an abstract second-order viscoelastic equation with infinite memory, time-varying delay and a nonlinear source term. To the best of our knowledge, there is no blow-up result of solutions for the nonlinear viscoelastic equation with time-varying delay. Moreover, our result extends the blow-up result obtained for problems without memory term to those with infinite memory and problems with constant delay to time-varying delay. Under suitable conditions on initial data, the kernel memory function and the involved functionals, we establish the local existence using the semigroup theory and we prove a finite time blow-up result for the solution with positive initial energy. We also give some applications to illustrate our main result. [ABSTRACT FROM AUTHOR]