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In this article, we find a lower bound for the blow-up time of solutions to some nonlinear parabolic equations under Robin boundary conditions in bounded domains of Rn  

This letter is concerned with the blow-up of the solutions to a semilinear parabolic problem with a reaction given by a variable exponent. Lower bounds for the time of blow-up are derived if the solutions blow up  

This paper deals with the blow-up of solutions to some nonlinear divergence form parabolic equations with nonlinear boundary conditions. We obtain a lower bound for the blow-up time of solutions in a bounded domain Ω⊆Rn, n≥. 3. L'article traite d'un problème d'explosion des solutions d'équations paraboliques non linéaires sous forme de divergence, avec des conditions aux limites non linéaires. On obtient une estimation d'une borne inférieure du temps d'explosion des solutions dans le cas d'un domaine borné Ω⊆Rn, n≥. 3  

We consider the chemotaxis system:. {ut=δu-∇;{dot operator}(uχ(v)∇;v)+f(u),x∈Ω,t>0,vt=δv-v+ug(u),x∈Ω,t>0, under homogeneous Neumann boundary conditions in a bounded domain Ω⊂Rn, n≥ 1, with smooth boundary and function f is assumed to generalize the logistic source:. f(u)=au-bu2,u≥0, with a>0,b>0. Moreover, χ( s) and g( s) are nonnegative smooth functions and satisfy:. χ(s)≤κ script(1+θ symbols)k,s≥0, with some κ script>0,θ symbol>0 and k>1,g(s)≤h0(1+hs)δ,s≥0,withh0>0,h≥0,δ≥0. We prove that for all positive values of κ script, a and b, classical solutions to the above system are uniformly-in-time bounded. This result extends a recent result by C. Mu, L. Wang, P. Zheng and Q. Zhang (2013)... 

This paper deals with the blow-up phenomena for a system of parabolic equations with nonlinear boundary conditions. We show that under some conditions on the nonlinearities, blow-up occurs at some finite time. We also obtain upper and lower bounds for the blow-up time when blow-up occurs. Copyright  

In this paper, we study the mathematical model proposed by Owen and Sherratt in 1997. We prove that the classical solutions to this model are uniformly-in-time bounded. © World Scientific Publishing Company  

We study the optimal value of p for solvability of the problem [Equation not available: see fulltext.]Here λ, α> 0 , p> 1 , f is a non-negative measurable function and [InlineEquation not available: see fulltext.], N⩾ 3 , is an open bounded domain with smooth boundary such that 0 ∈ Ω. We find the critical threshold exponent p+(λ, α) for solvability of (1) and show that if [InlineEquation not available: see fulltext.], 1 < p< p+(λ, α) and [InlineEquation not available: see fulltext.] for some sufficiently small c0> 0 , then there exists a solution as a limit of solutions to approximating problems. Moreover, for p⩾ p+(λ, α) we show that a complete blow-up phenomenon occurs. © 2017, Springer... 

The existence of structure for ionizing fast-strong and fast-weak detonation waves in magnetohydrodynamics are proved. The reactions are assumed to be one step exothermic reactions with a natural discontinuous reaction rate function. The problem is studied for a general gas, considering some general thermodynamics rules which described by a fairly mild set of hypotheses. The uniqueness and nonuniqueness of structure are also considered. © 2007 Elsevier Ltd. All rights reserved  

In this paper, we study the existence of a positive solution to the elliptic problem: (Formula presented.) Here (Formula presented.) (N>2s) is an open bounded domain with smooth boundary, (Formula presented.) and (Formula presented.). For (Formula presented.), we take advantage of the convexity of Ω. The operator (Formula presented.) indicates the restricted fractional Laplacian, and μ is a non-negative Radon measure as a source term. The assumptions on f and h will be precised later. Besides, we will discuss the notion of entropy solution and its uniqueness for some specific measures. © 2020, © 2020 Informa UK Limited, trading as Taylor & Francis Group  

This paper deals with the question of existence of periodic solutions of nonautonomous predator-prey dynamical systems with Beddington-DeAngelis functional response. We explore the periodicity of this system on time scales. New sufficient conditions are derived for the existence of periodic solutions. These conditions extend previous results presented in [M. Bohner, M. Fan, J. Zhang, Existence of periodic solutions in predator-prey and competition dynamic systems, Nonlinear. Anal.: Real World Appl. 7 (2006) 1193-1204; M. Fan, Y. Kuang, Dynamics of a nonautonomous predator-prey system with the Beddington-DeAngelies functional response, J. Math. Anal. Appl. 295 (2004) 15-39; J. Zhang, J. Wang,... 

In this paper we explore the existence of periodic solutions of a nonautonomous semi-ratio-dependent predator-prey dynamical system with functional responses on time scales. To illustrate the utility of this work, we should mention that, in our results this system with a large class of monotone functional responses, always has at least one periodic solution. For instance, this system with some celebrated functional responses such as Holling type-II (or Michaelis-Menten), Holling type-III, Ivlev, mx (Holling type I), sigmoidal [e.g., Real and mx2/((A + x)(B +x))] and some other monotone functions, has always at least one ω-periodic solution. Besides, for some well-known functional responses... 

In this Note, sharp sufficient conditions for the existence of periodic solutions of a nonautonomous discrete time semi-ratio-dependent predator-prey system with functional responses are derived. In our results this system with any monotone functional response bounded by polynomials in R+, always has at least one ω-periodic solution. In particular, this system with the most popular functional responses Michaelis-Menten, Holling type-II and III, sigmoidal, Ivlev and some other monotone response functions, always has at least one ω-periodic solution. To cite this article: M. Fazly, M. Hesaaraki, C. R. Acad. Sci. Paris, Ser. I 345 (2007). © 2007 Académie des sciences  

In this note, we study on the existence and uniqueness of a positive solution to the following doubly singular fractional problem: {(-Δ)su=K(x)uq+f(x)uγ+μinΩ,u>0inΩ,u=0in(RN\u03a9).Here Ω ⊂ RN (N> 2 s) is an open bounded domain with smooth boundary, s∈ (0 , 1) , q> 0 , γ> 0 , and K(x) is a positive Hölder continuous function in which behaves as dist (x, ∂Ω) -β near the boundary with 0 ≤ β< 2 s. Also, 0 ≤ f, μ∈ L1(Ω) , or non-negative bounded Radon measures in Ω. Moreover, we assume that 0<βs+q<1, or βs+q>1 with 2 β+ q(2 s- 1) < (2 s+ 1). For s∈(0,12), we take advantage of the convexity of Ω. For any γ> 0 , we will prove the existence of a positive weak (distributional) solution to the above... 

In this paper, we study the existence of a positive solution to the elliptic problem: (Formula presented.) Here (Formula presented.) (N>2s) is an open bounded domain with smooth boundary, (Formula presented.) and (Formula presented.). For (Formula presented.), we take advantage of the convexity of Ω. The operator (Formula presented.) indicates the restricted fractional Laplacian, and μ is a non-negative Radon measure as a source term. The assumptions on f and h will be precised later. Besides, we will discuss the notion of entropy solution and its uniqueness for some specific measures. © 2020 Informa UK Limited, trading as Taylor & Francis Group  

Dengue is among the most important infectious diseases in the world. The main contribution of our paper is to present a mixed system of partial and ordinary differential equations. This combined model is a generalization of the two presented mathematical models (A. L. de Araujo, J. L. Boldrini and B. M. Calsavara, An analysis of a mathematical model describing the geographic spread of dengue disease, J. Math. Anal. Appl. 444 (2016) 298-325) and (L. Cai, X. Li, N. Tuncer, M. Martcheva and A. A. Lashari, Optimal control of a malaria model with asymptomatic class and superinfection, Math. Biosci. 288 (2017) 94-108), describing the geographic spread of dengue disease. Our model has the ability... 

We use the singular sources method to detect the shape of the obstacle in a mixed boundary value problem. The basic idea of the method is based on the singular behavior of the scattered field of the incident point-sources on the boundary of the obstacle. Moreover we take advantage of the scattered field estimate by the backprojection operator. Also we give a uniqueness proof for the shape reconstruction. © 2004 Elsevier Inc. All rights reserved  

Ionizing shock waves in magnetofluiddynamics occur when the coefficient of electrical conductivity is very small ahead of the shock and very large behind it. For planner motion of plasma, the structure of such shock waves are stated in terms of a system of four-dimensional equations. In this paper, we show that for the above electrical conductivity as well as for limiting cases, that is, when this coefficient is zero ahead of the shock and/or is infinity behind it, ionizing fast, slow, switch-on and switch-off shocks admit structure. This means that physically these shocks occur. © 2002 Hindawi Publishing Corporation. All rights reserved  

This paper is concerned with the existence of positive solution to a class of singular fourth order elliptic equation of Kirchhoff type (Formula Presented.)▵2u-λM(‖∇u‖2)▵u-μ|x|4u=h(x)uγ+k(x)uα,under Navier boundary conditions, u= ▵u= 0. Here Ω⊂ RN, N≥ 1 is a bounded C4-domain, 0 ∈ Ω, h(x) and k(x) are positive continuous functions, γ∈ (0 , 1) , α∈ (0 , 1) and M: R+→ R+ is a continuous function. By using Galerkin method and sharp angle lemma, we will show that this problem has a positive solution for m0 and 0 < μ< μ∗. Here μ∗=(N(N-4)4)2 is the best constant in the Hardy inequality. Besides, if μ= 0 , λ> 0 and h, k are Lipschitz functions, we show that this problem has a positive smooth... 

In this paper, we study the effect of Hardy potential on the existence or nonexistence of solutions to the following fractional problem involving a singular nonlinearity: {(−Δ)su=λu|x|2s+μuγ+fin Ω,u>0in Ω,u=0in (RN∖Ω). Here 0 < s< 1 , λ> 0 , γ> 0 , and Ω ⊂ RN (N> 2 s) is a bounded smooth domain such that 0 ∈ Ω. Moreover, 0 ≤ μ, f∈ L1(Ω). For 0 < λ≤ Λ N,s, Λ N,s being the best constant in the fractional Hardy inequality, we find a necessary and sufficient condition for the existence of a positive weak solution to the problem with respect to the data μ and f. Also, for a regular datum of f, under suitable assumptions, we obtain some existence and uniqueness results and calculate the rate of...