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We consider the parabolic system u t − a Δ u = f (v) , v t − b Δ v = g (u) in Ω × (0 , T) , where a , b > 0 , f , g : [ 0 , ∞) → [ 0 , ∞) are non-decreasing continuous functions and either Ω is a bounded domain with smooth boundary ∂Ω or the whole space R N. We characterize the functions f and g so that the system has a local solution for every initial data (u 0 , v 0) ∈ L r (Ω) × L s (Ω) , u 0 , v 0 ≥ 0 , r , s ∈ [ 1 , ∞). [ABSTRACT FROM AUTHOR]

In this paper, we consider the Cauchy problem of a viscous compressible shallow water equations with the Coriolis force term and non-constant viscosities. More precisely, the viscous coefficients are constants multiple of height, the equations are degenerate when vacuum appears. For initial data allowing vacuum, the local existence of strong solution is obtained and a blow-up criterion is established. [ABSTRACT FROM AUTHOR]

Abstract We begin by proving a local existence result for a fractional Caputo nonlocal thermistor problem. Then additional existence and continuation theorems are obtained, ensuring global existence of solutions.

We study the density-dependent incompressible Cahn–Hilliard–Navier–Stokes system, which describes a two-phase flow of two incompressible fluids with different densities. We establish the local existence and uniqueness of strong solutions to the initial value problem in a bounded domain, when the initial density function enjoys a positive lower bound. [ABSTRACT FROM AUTHOR]

In his monograph "Systèmes Orthogonaux" (Darboux, in Leçons sur les systèmes orthogonaux et les coordonnées curvilignes, Gauthier-Villars, Paris, 1910) Darboux stated three theorems providing local existence and uniqueness of solutions to first-order systems of the type ∂ x i u α (x) = f i α (x , u (x)) , i ∈ I α ⊆ { 1 , ⋯ , n }. For a given point x ¯ ∈ R n it is assumed that the values of the unknown u α are given locally near x ¯ along { x | x i = x ¯ i for each i ∈ I α } . The more general of the theorems, Théorème III, was proved by Darboux only for the cases n = 2 and 3. In this work we formulate and prove a generalization of Darboux's Théorème III which applies to systems of the form r i (u α) | x = f i α (x , u (x)) , i ∈ I α ⊆ { 1 , ⋯ , n } where R = { r i } i = 1 n is a fixed local frame of vector fields near x ¯ . The data for u α are prescribed along a manifold Ξ α containing x ¯ and transverse to the vector fields { r i | i ∈ I α } . We identify a certain Stable Configuration Condition (SCC). This is a geometric condition that depends on both the frame R and on the manifolds Ξ α ; it is automatically met in the case considered by Darboux. Assuming the SCC and the relevant integrability conditions are satisfied, we establish local existence and uniqueness of a C 1 -solution via Picard iteration for any number of independent variables n. [ABSTRACT FROM AUTHOR]

In this paper, we study traveling wave solutions of the chemotaxis system {ut Δ u − χ1∇(u∇v1) + χ2∇(u∇v2) + u(a−bu), x ∈ R τ∂tv1 = (Δ − λ1I)v1+μ1u, x ∈ R, τ∂v2 = (Δ − λ2I)v2 + μ2u, x ∈ R,(0.1) where τ > 0, χi > 0,λi > 0, μi > 0 (i = 1,2) and a > 0, b > 0 are constants. Under some appropriate conditions on the parameters, we show that there exist two positive constant c∗(τ,χ1,μ1,λ1,χ2,μ2,λ2) < c∗∗ (τ,χ1,μ1,λ1,χ2,μ2,λ2) such that for every c∗(τ,χ1,μ1,λ1,χ2,μ2,λ2) ≤ c < c∗∗ (τ,χ1,μ1,λ1,χ2,μ2,λ2), (0.1) has a traveling wave solution (u,v1,v2)(x,t) = (U,V1,V2) (x − ct) connecting (ab,aμ1bλ1,aμ2bλ2) and (0,0,0) satisfying limz→∞U(z)e−μz=1, where μ∈(0,√a) is such that c = cμ: = μ + aμ . Moreover, lim(χ1,χ2) → (0+,0+))c∗∗(τ,χ1,μ1,λ1,χ2,μ2,λ2) = ∞ and lim(χ1,χ2) → (0+,0+))c∗(τ,χ1,μ1,λ1,χ2,μ2,λ2) = c~μ∗, where ~μ∗=min{√a,√λ1+τa(1−τ)+,√λ2+τa(1−τ)+} . We also show that (1) has no traveling wave solution connecting (ab,aμ1bλ1,aμ2bλ2) and (0,0,0) with speed c<2√a. [ABSTRACT FROM AUTHOR]

In this paper, we study the blow-up, and global existence of solutions to the following time–space fractional diffusion problem 0 C D t α u (x , t) + (− Δ) β ∕ 2 u (x , t) = 0 J t γ | u | p − 1 u (x , t) , x ∈ R N , t > 0 , u (x , 0) = u 0 (x) ∈ C 0 (R N) , where 0 < α < 1 − γ , 0 < γ < 1 , 0 < β ≤ 2 , p > 1 , 0 J t γ denotes the left Riemann–Liouville fractional integral of order γ , 0 C D t α is the Caputo fractional derivative of order α and (− Δ) β ∕ 2 stands for the fractional Laplacian operator of order β ∕ 2. We show that if p < 1 + β (α + γ) ∕ α N , then every nonnegative solution blows up in finite time, and if p ≥ 1 + β (α + γ) ∕ α N and ‖ u 0 ‖ L q c (R N) is sufficiently small, where q c = α N (p − 1) ∕ β (α + γ) , then the problem has global solutions. Finally, we give an upper bound estimate of the life span of blowing-up solutions. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following chemotaxis systems of parabolic–elliptic–elliptic type on R N , u t = Δ u - χ 1 ∇ (u ∇ v 1) + χ 2 ∇ (u ∇ v 2) + u (a - b u) , x ∈ R N , t > 0 , 0 = (Δ - λ 1 I) v 1 + μ 1 u , x ∈ R N , t > 0 , 0 = (Δ - λ 2 I) v 2 + μ 2 u , x ∈ R N , t > 0 , u (· , 0) = u 0 , x ∈ R N , where χ i ≥ 0 , λ i > 0 , μ i > 0 ( i = 1 , 2 ) and a > 0 , b > 0 are constant real numbers, and N is a positive integer. First, under some conditions on the parameters χ i , μ i , λ i , a , b and N, we prove the global existence and boundedness of classical solutions (u (x , t ; u 0) , v 1 (x , t ; u 0) , v 2 (x , t ; u 0)) for nonnegative, bounded, and uniformly continuous initial functions u 0 (x) . Next, we explore the asymptotic stability of the constant equilibrium (a b , μ 1 λ 1 a b , μ 2 λ 2 a b) and prove under some further assumption on the parameters that, for every strictly positive initial u 0 (x) , lim t → ∞ ‖ u (· , t ; u 0) - a b ‖ ∞ + ‖ λ 1 v 1 (· , t ; u 0) - a b μ 1 ‖ ∞ + ‖ λ 2 v 2 (· , t ; u 0) - a b μ 2 ‖ ∞ = 0. Finally, we investigate the spreading properties of the global solutions with compactly supported initial functions. We show that under some conditions on the parameters, there are two positive numbers 0 < c - ∗ (χ 1 , μ 1 , λ 1 , χ 2 , μ 2 , λ 2) ≤ c + ∗ (χ 1 , μ 1 , λ 1 , χ 2 , μ 2 , λ 2) such that for every nonnegative initial function u 0 (x) with nonempty and compact support, we have lim t → ∞ sup | x | ≤ c t | u (x , t ; u 0) - a b | + sup | x | ≤ c t | λ 1 v 1 (x , t ; u 0) - a b μ 1 | + sup | x | ≤ c t | λ 2 v 2 (x , t ; u 0) - a b μ 2 | = 0 whenever 0 ≤ c < c - ∗ (χ 1 , μ 1 , λ 1 , χ 2 , μ 2 , λ 2) , and lim t → ∞ sup | x | ≥ c t | u (x , t ; u 0) | + sup | x | ≥ c t | v 1 (x , t ; u 0) | + sup | x | ≥ c t | v 2 (x , t ; u 0) | = 0 whenever c > c + ∗ (χ 1 , μ 1 , λ 1 , χ 2 , μ 2 , λ 2) . Furthermore we show that lim (χ 1 , χ 2) → (0 , 0) c - ∗ (χ 1 , μ 1 , λ 1 , χ 2 , μ 2 , λ 2) = lim (χ 1 , χ 2) → (0 , 0) c + ∗ (χ 1 , μ 1 , λ 1 , χ 2 , μ 2 , λ 2) = 2 a. [ABSTRACT FROM AUTHOR]

Local and blowing‐up solutions for the Cauchy problem for a system of space and time fractional evolution equations with time‐nonlocal nonlinearities of exponential growth are considered. The existence and uniqueness of the local mild solution is assured by the Banach fixed point principle. Then, we establish a blow‐up result by Pokhozhaev capacity method. Finally, under some suitable conditions, an estimate of the life span of blowing‐up solutions is established. [ABSTRACT FROM AUTHOR]

3D stochastic Euler equations with a special form of multiplicative noise are considered. A Constantin-Iyer type representation in Euler-Lagrangian form is given, based on stochastic characteristics. Local existence and uniqueness of solutions in suitable Hölder spaces is proved from the Euler-Lagrangian formulation. [ABSTRACT FROM AUTHOR]

Abstract This article focuses on a quasilinear wave equation of p -Laplacian type: u t t − Δ p u − Δ u t = f (u) in a bounded domain Ω ⊂ R 3 with a sufficiently smooth boundary Γ = ∂ Ω subject to a generalized Robin boundary condition featuring boundary damping and a nonlinear source term. The operator Δ p , 2 < p < 3 , denotes the classical p -Laplacian. The interior and boundary terms f (u) , h (u) are sources that are allowed to have a supercritical exponent, in the sense that their associated Nemytskii operators are not locally Lipschitz from W 1 , p (Ω) into L 2 (Ω) or L 2 (Γ). Under suitable assumptions on the parameters we provide a rigorous proof of existence of a local weak solution which can be extended globally in time, provided the damping terms dominate the corresponding sources in an appropriate sense. Moreover, a blow-up result is proved for solutions with negative initial total energy. [ABSTRACT FROM AUTHOR]

In this paper we consider a porous-elastic system consisting of nonlinear boundary/interior damping and nonlinear boundary/interior sources. Our interest lies in the theoretical understanding of the existence, finite time blow-up of solutions and their exponential decay using non-trivial adaptations of well-known techniques. First, we apply the conventional Faedo-Galerkin method with standard arguments of density on the regularity of initial conditions to establish two local existence theorems of weak solutions. Moreover, we detail the uniqueness result in some specific cases. In the second theme, we prove that any weak solution possessing negative initial energy has the latent blow-up in finite time. Finally, we obtain the so-called exponential decay estimates for the global solution under the construction of a suitable Lyapunov functional. In order to corroborate our theoretical decay, a numerical example is provided. [ABSTRACT FROM AUTHOR]

Abstract In atmospheric science, slice models are frequently used to study the behaviour of weather, and specifically the formation of atmospheric fronts, whose prediction is fundamental in meteorology. In 2013, Cotter and Holm introduced a new slice model, which they formulated using Hamilton's variational principle, modified for this purpose. In this paper, we show the local existence and uniqueness of strong solutions of the related ISM (Incompressible Slice Model). The ISM is a modified version of the Cotter–Holm Slice Model (CHSM) that we have obtained by adapting the Lagrangian function in Hamilton's principle for CHSM to the Euler–Boussinesq Eady incompressible case. Besides proving local existence and uniqueness, in this paper we also construct a blow-up criterion for the ISM, and study Arnold's stability around a restricted class of equilibrium solutions. These results establish the potential applicability of the ISM equations in physically meaningful situations. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with local existence and blowup of a unique solution to a time‐space fractional evolution equation with a time nonlocal nonlinearity of exponential growth. At first, we prove the existence and uniqueness of the local solution by the Banach contraction mapping principle. Then, the blowup result of the solution in finite time is established by the test function method with a judicious choice of the test function. [ABSTRACT FROM AUTHOR]

The skew mean curvature flow (SMCF), which origins from the study of fluid dynamics, describes the evolution of a codimension two submanifold along its binormal direction. We study the basic properties of the SMCF and prove the existence of a short-time solution to the initial value problem of the SMCF of compact surfaces in Euclidean space ℝ 4 . A Sobolev-type embedding theorem for the second fundamental forms of two-dimensional surfaces is also proved, which might be of independent interest. [ABSTRACT FROM AUTHOR]

In the present paper we consider initial value problems for Hilfer fractional differential equations and for system of Hilfer fractional differential equations. By using equivalent integral equations and some fixed point theorems, we study the local existence of solutions. We extend these local existence results globally with the help of continuation theorems and generalized Gronwall inequality. [ABSTRACT FROM AUTHOR]

We consider the system ut=Δu−χ∇·uv∇v−uv+B1vt=Δv−v+uv+B2of partial differential equations for some positive χ and some Ω⊂Rn, n ≥ 2. While there are results regarding the existence of global solutions in one dimension for any χ > 0, there are not yet any for the higher‐dimensional case. This work in which we establish global solutions for any χ∈0,2n is a first step toward filling this gap. [ABSTRACT FROM AUTHOR]

In this article we study the asymptotic behavior of solutions, in the sense of pullback attractors, of the evolution system {utt+Δ²u+a(t)Δ=f(t,u),t--a(t)ut=0,t>t, xO,t>t, xO, subject to boundary conditions u=u==0, t>t, xO, where O is a bounded domain in RN with N=2, which boundary O is assumed to be a C4-hypersurface, κ>0 is constant, a is an Hölder continuous function and f is a dissipative nonlinearity locally Lipschitz in the second variable. Using the theory of uniform sectorial operators, in the sense of P. Sobolevskii ([23]), we give a partial description of the fractional power spaces scale for the thermoelastic plate operator and we show the local and global well-posedness of this non-autonomous problem. Furthermore we prove existence and uniform boundedness of pullback attractors. [ABSTRACT FROM AUTHOR]

In this paper, we consider an initial boundary value problem for nonlinear Love equation with infinite memory. By combining the linearization method, the Faedo–Galerkin method, and the weak compactness method, the local existence and uniqueness of weak solution is proved. Using the potential well method, it is shown that the solution for a class of Love-equation exists globally under some conditions on the initial datum and kernel function.

We consider the motion of viscous compressible magnetohydrodynamics fluid in a domain bounded by a free surface. In the external domain, there is electromagnetic field generated by some currents that keeps the magnetohydrodynamics flow in the bounded domain. Then on the free surface, transmission conditions for electromagnetic fields are imposed. In this paper, we prove the existence of local regular solutions by the method of successive approximations. The L2 approach is used. This helps us to treat the transmission conditions. [ABSTRACT FROM AUTHOR]

This article focuses on a quasilinear wave equation of p-Laplacian type: utt - Δpu - Δ ut = 0 in a bounded domain Ω C R³ with a sufficiently smooth boundary condition featuring boundary damping and a nonlinear source term. The operator Δp, 2 < p < 3, denotes the classical p-Laplacian. The nonlinear boundary term f(u) is a source feedback that is allowed to have a supercritical exponent, in the sense that the associated Nemytskii operator is not locally Lipschitz from W1,p(Ω) into L²(Γ). Under suitable assumptions on the parameters we provide a rigorous proof of existence of a local weak solution which can be extended globally in time provided the source term satisfies an appropriate growth condition. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem for the higher-order nonlinear Schrödinger equation ... where k, p ∈ N, k ≥ 2, λ ∈ C. We prove local existence of solutions for the case of singular initial data u0 (x) including the Dirac delta function. [ABSTRACT FROM AUTHOR]

We establish a priori estimates for local-in-time existence of solutions to the water wave model consisting of the 3D incompressible Euler equations on a domain with a free surface, without surface tension, under minimal regularity assumptions on the initial data and the Rayleigh-Taylor sign condition. The initial data are allowed to be rotational and they are assumed to belong to H2.5+δ, where δ > 0 is arbitrary. [ABSTRACT FROM AUTHOR]

We review recent progress on the long-time regularity of solutions of the Cauchy problem for the water waves equations, in two and three dimensions. We begin by introducing the free boundary Euler equations and discussing the local existence of solutions using the paradifferential approach.We then describe in a unified framework, using the Eulerian formulation, global existence results for three- and two-dimensional gravity waves, and our joint result (with Deng and Pausader) on global regularity for the gravity-capillary model in three dimensions. We conclude this review with a short discussion about the formation of singularities and give a few additional references to other interesting topics in the theory. [ABSTRACT FROM AUTHOR]

We study the Cauchy problem for the semilinear fractional heat equation ut = △α/2u + f(u) with non-negative initial value u0 ∈ Lq(ℝn) and locally Lipschitz, non-decreasing, non-negative source term f. For f satisfying the Osgood-type condition ∫ 1 ∞ d s f ( s ) = ∞ $\begin {array}{}\displaystyle\int_{1}^{\infty}\frac{ds}{f(s)}=\infty\end {array}$ , we show that there exist initial conditions such that the equation has no local solution in L l o c 1 ( R n ) $\begin {array}{}L^{1}_{loc}(\mathbb{R}^{n})\end {array}$. [ABSTRACT FROM AUTHOR]

We address the local existence and uniqueness of solutions for the 3D Euler equations with a free interface. We prove the local well-posedness in the rotational case when the initial datum $$u_0$$ satisfies $$u_0\in H^{2.5+\delta }$$ and , where $$\delta >0$$ is arbitrarily small, under the Taylor condition on the pressure. [ABSTRACT FROM AUTHOR]

We consider the Muskat Problem with surface tension in two dimensions over the real line, with H initial data and allowing the two fluids to have different constant densities and viscosities. We take the angle between the interface and the horizontal, and derive an evolution equation for it. Via energy methods, it has been shown that a unique solution $${\theta}$$ exists locally and can be continued while $${||\theta||_{s}}$$ remains bounded and the arc chord condition holds. We prove that when both fluids have the same viscosity and the initial data is sufficiently small, the energy estimate is dominated by second-order dissipative terms. As a result, the energy is non-increasing, and that the resulting solution $${\theta}$$ exists globally in time. [ABSTRACT FROM AUTHOR]

In this study, we consider an extended attraction two species chemotaxis system of parabolic-parabolic-elliptic type with nonlocal terms under homogeneous Neuman boundary conditions in a bounded domain Ω ⊂ ℝn(n ≤1) with smooth boundary. We first prove the global existence of nonnegative classical solutions for various explicit parameter regions. Next, under some further explicit conditions on the coeficients and on the chemotaxis sensitivities, we show that the system has a unique positive constant steady state solution which is globally asymptotically stable. Finally, we also find some explicit conditions on the coeficient and on the chemotaxis sensitivities for which the phenomenon of competitive exclusion occurs in the sense that as time goes to infinity, one of the species dies out and the other reaches its carrying capacity. The method of eventual comparison is used to study the asymptotic behavior. [ABSTRACT FROM AUTHOR]

We begin by proving a local existence result for a fractional Caputo nonlocal thermistor problem. Then additional existence and continuation theorems are obtained, ensuring global existence of solutions. [ABSTRACT FROM AUTHOR]