Search

It is well known from the work of [2] that the Fujita phenomenon for reaction-diffusion evolution equations with power nonlinearities does not occur on the hyperbolic space $\mathbb{H}^N$, thus marking a striking difference with the Euclidean situation. We show that, on classes of manifolds in which the bottom $\Lambda$ of the $L^2$ spectrum of $-\Delta$ is strictly positive (the hyperbolic space being thus included), a different version of the Fujita phenomenon occurs for other kinds of nonlinearities, in which the role of the critical Fujita exponent in the Euclidean case is taken by $\Lambda$. Such nonlinearities are time-independent, in contrast to the ones studied in [2]. As a consequence of our results we show that, on a class of manifolds much larger than the case $M=\mathbb{H}^N$ considered in [2], solutions to (1.1) with power nonlinearity $f(u)=u^p$, $p>1$, and corresponding to sufficiently small data, are global in time., Comment: To appear in "Annali di Matematica Pura e Applicata"

This paper particularly addresses and discusses some analytical studies on the existence and uniqueness of global or blow-up solutions under the traveling profile forms for a free boundary problem of two-dimensional diffusion equations of moving fractional order. It does so by applying the properties of Schauder's and Banach's fixed point theorems. For application purposes, some examples of explicit solutions are provided to demonstrate the usefulness of our main results.

We consider reaction-diffusion equations driven by the $p$-Laplacian on noncompact, infinite volume manifolds assumed to support the Sobolev inequality and, in some cases, to have $L^2$ spectrum bounded away from zero, the main example we have in mind being the hyperbolic space of any dimension. It is shown that, under appropriate conditions on the parameters involved and smallness conditions on the initial data, global in time solutions exist and suitable smoothing effects, namely explicit bounds on the $L^\infty$ norm of solutions at all positive times, in terms of $L^q$ norms of the data. The geometric setting discussed here requires significant modifications w.r.t. the Euclidean strategies., Comment: arXiv admin note: substantial text overlap with arXiv:2012.02084

This work deals with a system of Kirchhoff-type equations with viscoelastic term and logarithmic nonlinearity. First, we proved the global existence by combining the potential well method with Faedo-Galerkin's procedure. Later, we established decay rate estimates by using multiplier method.

In this paper, we study an anomalous pseudo-parabolic Kirchhoff-type dynamical model aiming to reveal the control problem of the initial data on the dynamical behavior of the solution in dynamic control system. Firstly, the local existence of solution is obtained by employing the Contraction Mapping Principle. Then, we get the global existence of solution, long time behavior of global solution and blowup solution for J(u 0) ⩽ d, respectively. In particular, the lower and upper bound estimates of the blowup time are given for J(u 0)

In this paper, we discuss global existence, boundness, blow-up and extinction properties of solutions for the Dirichlet boundary value problem of the $ p $-Laplacian equations with logarithmic nonlinearity $ u_{t}-{\rm{div}}(|\nabla u|^{p-2}\nabla u)+\beta|u|^{q-2}u = \lambda |u|^{r-2}u\ln{|u|} $, where $ 1 < p < 2 $, $ 1 < q\leq2 $, $ r > 1 $, $ \beta, \lambda > 0 $. Under some appropriate conditions, we obtain the global existence of solutions by means of the Galerkin approximations, then we prove that weak solution is globally bounded and blows up at positive infinity by virtue of potential well theory and the Nehari manifold. Moreover, we obtain the decay estimate and the extinction of solutions.

We provide a short review of existing models with multiple taxis performed by (at least) one species and consider a new mathematical model for tumor invasion featuring two mutually exclusive cell phenotypes (migrating and proliferating). The migrating cells perform nonlinear diffusion and two types of taxis in response to non-diffusing cues: away from proliferating cells and up the gradient of surrounding tissue. Transitions between the two cell subpopulations are influenced by subcellular (receptor binding) dynamics, thus conferring the setting a multiscale character. We prove global existence of weak solutions to a simplified model version and perform numerical simulations for the full setting under several phenotype switching and motility scenarios. We also compare (via simulations) this model with the corresponding haptotaxis-chemotaxis one featuring indirect chemorepellent production and provide a discussion about possible model extensions and mathematical challenges., Comment: 43 pages, 15 figures

We consider strong damped wave equation involving the fractional Laplacian with nonlinear source. The results of global solution under necessary conditions on the critical exponent are established. The existence is proved by using the Galerkin approximations combined with the potential well theory. Moreover, we showed new decay estimates of global solution.

This paper is concerned with the existence of a compact global attractor for a class of competition model in n−dimensional (n ≥ 1) domains. Using mathematical induction and more detailed interpolation estimates, especially Gagliardo-Nirenberg inequality, we obtain the existence of a compact global attractor, which implies the uniform boundedness of the global solutions. In particular, we get that the Shigesada-Kawasaki-Teramoto competition model has a compact global attractor for n < 10. The result of the S-K-T model extends the existence results of compact global attractor in [21] from n < 8 to n < 10, and extends the uniform boundedness results of the global solutions in [17] to the non-convex domain.

This paper is devoted to the Schrödinger-Choquard equation with linear damping. Global existence and scattering are proved depending on the size of the damping coefficient.

We derive a blow-up dichotomy for positive solutions of fractional semilinear heat equations on the whole space. That is, within a certain class of convex source terms, we establish a necessary and sufficient condition on the source for all positive solutions to become unbounded in finite time. Moreover, we show that this condition is equivalent to blow-up of all positive solutions of a closely-related scalar ordinary differential equation., Comment: 19 pages

A time-fractional space-nonlocal reaction-diffusion equation in a bounded domain is considered. First, the existence of a unique local mild solution is proved. Applying Poincaré inequality it is obtained the existence and boundedness of global classical solution for small initial data. Under some conditions on the initial data, we show that solutions may experience blow-up in a finite time.

In this paper, we study the fractional p-Laplacian evolution equation with arbitrary initial energy, $$\begin{array}{} \displaystyle u_t(x,t) + (-{\it\Delta})_p^s u(x,t) = f(u(x,t)), \quad x\in {\it\Omega}, \,t \gt 0, \end{array} $$ where $\begin{array}{} (-{\it\Delta})_p^s \end{array} $ is the fractional p-Laplacian with $\begin{array}{} p \gt \max\{\frac{2N}{N+2s},1\} \end{array} $ and s ∈ (0, 1). Specifically, by the modified potential well method, we obtain the global existence, uniqueness, and blow-up in finite time of the weak solution for the low, critical and high initial energy cases respectively.

We introduce a generalized concept of solutions for reaction–diffusion systems and prove their global existence. The only restriction on the reaction function beyond regularity, quasipositivity and mass control is special in that it merely controls the growth of cross-absorptive terms. The result covers nonlinear diffusion and does not rely on an entropy estimate.

The initial value problem for incompressible Hookean viscoelastic motion in three space dimensions has global strong solutions with small displacements.

We study global in time existence versus blow-up in finite time of solutions to the Cauchy problem for the porous medium equation with a variable density $ρ(x)$ and a power-like reaction term posed in the one dimensional interval $(-R,R)$, $R>0$. Here the weight function is singular at the boundary of the domain $(-R,R)$, indeed it is such that $ρ(x)\sim (R-|x|)^{-q}$ as $|x|\to R$, with $q\ge0$. We show a different behavior of solutions depending on the three cases when $q>2$, $q=2$ and $q, arXiv admin note: text overlap with arXiv:2007.12005, arXiv:1911.06043, arXiv:1911.07305

In this paper, we consider a time fractional diffusion system with a nonlinear memory term in a bounded domain. We mainly prove some blow-up and global existence results for this problem. Moreover, we also give the decay estimates of the global solutions. Our proof relies on the eigenfunction method combined with the asymptotic behavior of the solution of a fractional differential inequality system, the estimates of the solution operators and the asymptotic behavior of the Mittag–Leffler function. In particular, we give the critical exponents of this problem in different cases. Our results show that, in some cases, whether one of the initial values is identically equal to zero has a great influence on blow-up and global existence of the solutions for this problem, which is a remarkable property of time fractional diffusion systems because the classical diffusion systems can not admit this property.

This note studies the linearly damped generalized Hartree equation iu˙−(−Δ)su+iau=±|u|p−2(Jγ∗|u|p)u,00,p≥2. Indeed, one proves an exponential scattering of the energy global solutions, with spherically symmetric datum. This means that, for large time, the solution goes exponentially to the solution of the associated free problem iu˙−(−Δ)su+iau=0, in Hs norm. The radial assumption avoids a loss of regularity in Strichartz estimates. The exponential scattering, which means that v:=eatu scatters in Hs, is proved in the energy sub-critical defocusing regime and in the mass-sub-critical focusing regime. This result is presented because of the gap due to the lack of scattering in the mass sub-critical regime, which seems not to be well understood. In this manuscript, one needs to overcome three technical difficulties which are mixed together: the first one is a fractional Laplace operator, the second one is a Choquard (non-local) source term, including the Hartree-type term when p=2 and the last one is a damping term iau. In a work in progress, the authors investigate the exponential scattering of global solutions to the above Schrödinger problem, with different kind of damping terms.

We consider a model from electro-magneto-hydrodynamics describing a plasma in bounded multi connected domains. A nontrivial solution exists for magnetic fields as the equilibrium of this model. Nonlinear stability of the nontrivial solution is proved based on time weighted maximal $$L_p$$ L p -regularity.

Global existence, positivity, uniform boundedness and extinction results of solutions to a system of reaction-diffusion equations on unbounded domain modeling two species on a predator–prey relationship is considered.

A Leslie–Gower predator–prey system with cross-diffusion subject to Neumann boundary conditions is considered. The global existence and boundedness of solutions are shown. Some sufficient conditions ensuring the existence of nonconstant solutions are obtained by means of the Leray–Schauder degree theory. The local and global stability of the positive constant steady-state solution are investigated via eigenvalue analysis and Lyapunov procedure. Based on center manifold reduction and normal form theory, Hopf bifurcation direction and the stability of bifurcating time-periodic solutions are investigated and a normal form of Bogdanov-Takens bifurcation is determined as well.

We consider the Cauchy problem for heat equation with fractional Laplacian and exponential nonlinearity. We establish local well-posedness result in Orlicz spaces. We derive the existence of global solutions for small initial data. We obtain decay estimates for large time in Lebesgue spaces.

Gliomas are primary brain tumors with a high invasive potential and infiltrative spread. Among them, glioblastoma multiforme (GBM) exhibits microvascular hyperplasia and pronounced necrosis triggered by hypoxia. Histological samples showing garland-like hypercellular structures (so-called pseudopalisades) centered around the occlusion site of a capillary are typical for GBM and hint on poor prognosis of patient survival. We propose a multiscale modeling approach in the kinetic theory of active particles framework and deduce by an upscaling process a reaction-diffusion model with repellent pH-taxis. We prove existence of a unique global bounded classical solution for a version of the obtained macroscopic system and investigate the asymptotic behavior of the solution. Moreover, we study two different types of scaling and compare the behavior of the obtained macroscopic PDEs by way of simulations. These show that patterns (not necessarily of Turing type), including pseudopalisades, can be formed for some parameter ranges, in accordance with the tumor grade. This is true when the PDEs are obtained via parabolic scaling (undirected tissue), while no such patterns are observed for the PDEs arising by a hyperbolic limit (directed tissue). This suggests that brain tissue might be undirected - at least as far as glioma migration is concerned. We also investigate two different ways of including cell level descriptions of response to hypoxia and the way they are related .

In this paper, we investigate the blow-up rate and global existence of solutions to a parabolic system with absorption and the free boundary. By using the comparison principle and super-sub solution method, we obtain some sufficient conditions on the global existence, blow-up in finite time of solutions, and blow-up sets when blow-up phenomenon occurs. Furthermore, the global solution is bounded and uniformly tends to zero, and it is either a global fast solution or a global slow solution. Finally, we obtain a trichotomy conclusion by considering the size of parameter σ.

In this paper, we investigate the blow-up rate and global existence of solutions to a parabolic system with absorption under the homogeneous Dirichlet boundary. By using the comparison principle and super-sub solution method, we obtain some sufficient conditions for the global existence and blow-up in finite time of solutions and establish some estimates of the upper and lower bounds of the blow-up rates. For the special case, if the domain is symmetric, for example, if it is a ball, the results of this paper also hold.

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.

In this paper, we study the initial value problem for infinite dimensional fractional non-autonomous reaction-diffusion equations. Applying general time-splitting methods, we prove the existence of solutions globally defined in time using convex sets as invariant regions. We expose examples, where biological and pattern formation systems, under suitable assumptions, achieve global existence. We also analyze the asymptotic behavior of solutions., 23 pages, 2 figures

WOS:000754310800004 In this work, we analyze the influence of the logarithmic source term on solutions to quasilinear riser equation. Firstly, we prove blow up results. Later, we obtain that solutions are global with negative initial energy.

We consider a hydrodynamic model of flocking-type with all-to-all interaction kernel in one-space dimension and establish global existence of entropy weak solutions with concentration to the Cauchy problem for any BV initial data that has finite total mass confined in a bounded interval and initial density uniformly positive therein. In addition, under a suitable condition on the initial data, we show that entropy weak solutions with concentration admit time-asymptotic flocking., Comment: 59 pages, 1 figure

We study existence and non-existence of global solutions to the semilinear heat equation with a drift term and a power-like source term \begin{document}$ u^p $\end{document}, on Cartan-Hadamard manifolds. Under suitable assumptions on Ricci and sectional curvatures, we show that, for any \begin{document}$ p>1 $\end{document}, global solutions cannot exists if the initial datum is large enough. Furthermore, under appropriate conditions on the drift term, global existence is obtained for any \begin{document}$ p>1 $\end{document}, if the initial datum is sufficiently small. We also deal with Riemannian manifolds whose Ricci curvature tends to zero at infinity sufficiently fast. We show that for any non trivial initial datum, for certain \begin{document}$ p $\end{document} depending on the Ricci curvature bound, global solutions cannot exist. On the other hand, for certain values of \begin{document}$ p $\end{document}, depending on the vector field \begin{document}$ b $\end{document}, global solutions exist, for sufficiently small initial data.

In this work we establish a weighted $L^2 - L^2$ estimate for inhomogeneous wave equation in 3-D, by introducing a Morawetz multiplier with weight to the power $s(1 \lt s \lt 2)$, and then integrating on the light cones and $t$ slice. With this weighted $L^2 - L^2$ estimate in hand, we may give a new proof of global existence for small data Cauchy problem of semilinear wave equation with supercritical power in 3-D. What is more, by combining the Huygens' principle for wave equations in 3-D, the global existence for semilinear wave equation with scale invariant damping in 3-D is established.

This paper deals with a class of Petrovsky system with nonlinear damping $$\begin{aligned} w_{tt}+\Delta _{\mathbb{B}}^{2}w-k_{2} \Delta _{\mathbb{B}}w_{t}+aw_{t} \vert w_{t} \vert ^{m-2}=bw \vert w \vert ^{p-2} \end{aligned}$$ w t t + Δ B 2 w − k 2 Δ B w t + a w t | w t | m − 2 = b w | w | p − 2 on a manifold with conical singularity, where $\Delta _{\mathbb{B}}$ Δ B is a Fuchsian-type Laplace operator with totally characteristic degeneracy on the boundary $x_{1}=0$ x 1 = 0 . We first prove the global existence of solutions under conditions without relation between m and p, and establish an exponential decay rate. Furthermore, we obtain a finite time blow-up result for local solutions with low initial energy $E(0)< d$ E ( 0 ) < d .

In this paper a class of nonlocal diffusion equations associated with a p-Laplace operator, usually referred to as p-Kirchhoff equations, are studied. By applying Galerkin’s approximation and the modified potential well method, we obtain a threshold result for the solutions to exist globally or to blow up in finite time for subcritical and critical initial energy. The decay rate of the L 2 norm is also obtained for global solutions. When the initial energy is supercritical, an abstract criterion is given for the solutions to exist globally or to blow up in finite time, in terms of two variational numbers. These generalize some recent results obtained in [Y. Han and Q. Li, Threshold results for the existence of global and blow-up solutions to Kirchhoff equations with arbitrary initial energy, Computers and Mathematics with Applications, 75(9):3283–3297, 2018].

In this paper, we deal with the existence, uniqueness, and finite time blow-up of the solution to the degenerate nonlinear parabolic problem: uτ=(ξrumuξ)ξ/ξr + up for 0 < ξ < a, 0 < τ < Γ, u (ξ, 0) = u0 (ξ) for 0 ≤ ξ ≤ a, and u (0, τ) = 0 = u (a, τ) for 0 < τ < Γ, where u0 (ξ) is a positive function and u0 (0) = 0 = u0 (a). In addition, we prove that u exists globally if a is small through constructing a global-exist upper solution, and uτ blows up in a finite time.

In this study we introduced and tested retarded conformable fractional integral inequalities utilizing non-integer order derivatives and integrals. In line with this purpose, we used the Katugampola type conformable fractional calculus which has several practical properties. These inequalities generalize some famous integral inequalities which provide explicit bounds on unknown functions. The results provided here had been implemented to the global existence of solutions to the conformable fractional differential equations with time delay.

In this article, we analyze the existence of Peregrine type solutions for the fractional reaction–diffusion equation by applying splitting-type methods. Peregrine type functions have two main characteristics, these are direct sum of functions of periodic type and functions that tend to zero at infinity. Well-posedness results are obtained for each particular characteristic, and for both combined. Fil: Besteiro, Agustin Tomas. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina Fil: Rial, Diego Fernando. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina

We construct the global existence and exponential time decay rates of mild solutions to the non-linear Fokker–Planck equation near a global Maxwellians with small-amplitude initial data in the low regularity function space Lk1LT∞Lv2 where the regularity assumption on the initial data is weaker.

The focus of the current paper is to investigate the initial boundary value problem for a system of viscoelastic wave equations of Kirchho type with a delay term in a bounded domain. At first,the energy decay rate is proved by Nakao's technique and expressed polynomially and exponentially depending on the parameter m. However and in the unstable set, for certain initial data, the blow-upof solutions is obtained.

The aim of this study is to construct the invariant regions in which we can establish the global existence of classical solutions for reaction-diffusion systems with a general full matrix of diffusion coefficients. Our techniques are based on invariant regions and Lyapunov functional methods. The nonlinear reaction term has been supposed to be of exponential growth.

In this note we establish exponentially fast smooth convergence for global curve diffusion flows, and discuss open problems relating embeddedness to global existence (Giga's conjecture) and the shape of Type I singularities (Chou's conjecture)., Comment: 11 pages

We consider reaction-diffusion equations either posed on Riemannian manifolds or in the Euclidean weighted setting, with pow\-er-type nonlinearity and slow diffusion of porous medium time. We consider the particularly delicate case $p, Comment: Final version. To appear in J. Math. Pures Appl

This paper is concerned with a chemotaxis system with singular sensitivity and logistic source, $$\begin{aligned} \textstyle\begin{cases} u_{t}=\Delta u-\chi _{1}\nabla \cdot (\frac{u}{w}\nabla w)+\mu _{1}u-\mu _{1}u^{\alpha }, &x\in \varOmega , t>0, \\ \upsilon _{t}=\Delta v-\chi _{2}\nabla \cdot ( \frac{\upsilon }{w}\nabla w)+\mu _{2}\upsilon -\mu _{2}\upsilon ^{\beta } , &x\in \varOmega , t>0, \\ w_{t}=\Delta w-(u+\upsilon )w, &x\in \varOmega , t>0, \end{cases}\displaystyle \end{aligned}$$ under the homogeneous Neumann boundary conditions and for widely arbitrary positive initial data in a bounded domain $\varOmega \subset \mathbb{R}^{n}\ (n\geq 1)$ with smooth boundary, where $\chi _{i}$ , $\mu _{i}>0$ $(i=1, 2)$ and α, $\beta >1$ . It is proved that there exists a global classical solution if $\max \{\chi _{1}, \chi _{2}\} \frac{n-2}{n}, \alpha =\beta =2$ for $n\geq 2$ or any $\chi _{i}>0$ $(i=1,2)$ , $\mu _{i}>0 $ $(i=1,2)$ , α, $\beta >1$ for $n=1$ .

We prove existence and boundedness of classical solutions for a family of viscous conservation laws in one space dimension for arbitrarily large time. The result relies on H. Amann's criterion for global existence of solutions and on suitable uniform-in-time estimates for the solution. We also apply J\"ungel's boundedness-by-entropy principle in order to obtain global existence for systems with possibly degenerate diffusion terms. This work is motivated by the study of a physical model for the space-time evolution of the strain and velocity of an anharmonic spring of finite length.

Canonical quantization of general relativity does not yield a unique quantum theory for gravity. This is in part due to operator ordering ambiguities. In this paper, we investigate the role of different operator orderings on the question of whether a big bang or big crunch singularity occurs. We do this in the context of the minisuperspace model of a Friedmann-Lemaitre-Robertson-Walker universe with Brown-Kuchar dust. We find that for a certain class of operator orderings such a singularity is eliminated without having to impose boundary conditions., 9 pages, no figures, Latex; v2 minor improvements

International audience; In this paper, we consider the focusing mass-critical nonlinear fourth-order Schrödinger equation. We prove that blowup solutions to this equation with initial data in $H^\gamma(\mathbb{R}^d), 5 \leq d \leq 7, \frac{56−3d+ \sqrt{137d^2 +1712d+3136}}{2(2d+32)} < \gamma < 2$ concentrate at least the mass of the ground state at the blowup time. This extends the work in [35] where Zhu-Yang-Zhang studied the formation of singularity for the equation with rough initial data in $\mathbb{R}^4$. We also prove that the equation is globally well-posed with initial data $u_0 \in H^\gamma(\mathbb{R}^d), 5 \leq d \leq 7, \frac{8d}{3d+8} < \gamma < 2$ satisfying $\|u_0\| _{L^2 (\mathbb{R}^d)} < \|Q\|_{L^2 (\mathbb{R}^d)}$, where $Q$ is the solution to the ground state equation.

In this paper, we consider the following Timoshenko system of thermo-viscoelasticity of type III with frictional damping and delay terms: { ρ 1 ⁢ φ t ⁢ t - K ⁢ ( φ x + ψ ) x = 0 , ( x , t ) ∈ ( 0 , 1 ) × ( 0 , ∞ ) , ρ 2 ⁢ ψ t ⁢ t - b ⁢ ψ x ⁢ x + K ⁢ ( φ x + ψ ) + β ⁢ θ x = 0 , ( x , t ) ∈ ( 0 , 1 ) × ( 0 , ∞ ) , ρ 3 ⁢ θ t ⁢ t - δ ⁢ θ x ⁢ x + γ ⁢ ψ t ⁢ t ⁢ x + ∫ 0 t g ⁢ ( t - s ) ⁢ θ x ⁢ x ⁢ ( s ) ⁢ d s + μ 1 ⁢ θ t ⁢ ( x , t ) + μ 2 ⁢ θ t ⁢ ( x , t - τ ) = 0 , ( x , t ) ∈ ( 0 , 1 ) × ( 0 , ∞ ) , \left\{\begin{aligned} &\displaystyle\rho_{1}\varphi_{tt}-K(\varphi_{x}+\psi)_% {x}=0,&&\displaystyle(x,t)\in(0,1)\times(0,\infty),\\ &\displaystyle\rho_{2}\psi_{tt}-b\psi_{xx}+K(\varphi_{x}+\psi)+\beta\theta_{x}% =0,&&\displaystyle(x,t)\in(0,1)\times(0,\infty),\\ &\displaystyle\rho_{3}\theta_{tt}-\delta\theta_{xx}+\gamma\psi_{ttx}+\int_{0}^% {t}g(t-s)\theta_{xx}(s)\,\mathrm{d}s+\mu_{1}\theta_{t}(x,t)+\mu_{2}\theta_{t}(% x,t-\tau)=0,&&\displaystyle(x,t)\in(0,1)\times(0,\infty),\end{aligned}\right. together with initial datum and boundary conditions of Dirichlet type, where g is a positive non-increasing relaxation function and μ 1 , μ 2 {\mu_{1},\mu_{2}} are positive constants. Under a hypothesis between the weight of the delay term and the weight of the friction damping term, we prove the global existence of solutions by using the Faedo–Galerkin approximations together with some energy estimates. Then, by introducing appropriate Lyapunov functionals, under the imposed constrain on the above two weights, we establish a general energy decay result from which the exponential and polynomial types of decay are only special cases.

International audience; We consider an optimal control problem of underground water contaminated by agricultural pollution. The economical intertemporal objective takes into account the trade-off between fertilizer use and cleaning costs. It is constrained by a hydrogeological model for the spread of the pollution in the aquifer. This model consists in a parabolic partial differential equation which is nonlinearly coupled through the dispersion tensor with an elliptic equation, in a three-dimensional domain. We prove the existence of a global optimal solution under various regularity assumptions and for a wide variety of boundary conditions. We also provide an asymptotic controllability result.

The main purpose of this paper is to study the global existence and uniqueness of solutions for three-dimensional incompressible magnetic induction equations with Hall effect provided that ∥u0∥H32+&epsilon, +∥b0∥H2(0&lt, &epsilon, &lt, 1) is sufficiently small. Moreover, using the Fourier splitting method and the properties of decay character r*, one also shows the algebraic decay rate of a higher order derivative of solutions to magnetic induction equations with the Hall effect.

This paper is concerned with semilinear wave equations with nonlinear interior and boundary sources and subject to a nonlinear dynamical boundary condition. By using the potential well method combined with a standard continuous argument, under appropriate assumptions imposed on the source term, we establish global existence of solutions. Moreover, for certain initial data in the unstable set, the finite time blow-up phenomenon is exhibited.

This paper is devoted to studying a semilinear parabolic equation with conical degeneration. First, we extend previous results on the vacuum isolating of solution with initial energy $J(u_0) \lt d$, where $d$ is the mountain pass level. Concretely, we obtain the explicit vacuum region, the global existence region and the blow-up region. Moreover, as far as the blow-up solution is concerned, we estimate the upper bound of the blow-up time and blow-up rate. Second, for all $p \gt 1$, we get a new sufficient condition, which demonstrates the finite time blow-up for arbitrary initial energy, and the upper bound estimate of blow-up time is obtained.