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In this paper, we study the solvability of one initial-boundary value problem for the Burgers equation with periodic boundary conditions in a nonlinearly degenerating domain. In this paper, we found an orthonormal basis for domains with time-varying boundaries. On this basis, we use the Faedo–Galerkin method to prove theorems about the unique solvability of the problem under consideration. We also present some numerical results in the form of graphs of solutions to the problem under study for various initial data. [ABSTRACT FROM AUTHOR]

We study some zero-flux attraction-repulsion chemotaxis models, with nonlinear production rates for the chemorepellent and the chemoattractant, whose formulation can be schematized as (⋄) $$\begin{equation} \begin{cases} u_t= \Delta u - \chi \nabla \cdot (u \nabla v)+\xi \nabla \cdot (u \nabla w) & {\rm in}\ \Omega \times (0,T_{\max}),\\ \tau v_t=\Delta v-\varphi(t,v)+f(u) & {\rm in}\ \Omega \times (0,T_{\max}),\\ \tau w= \Delta w - \psi(t,w) + g(u) & {\rm in}\ \Omega \times (0,T_{\max}). \end{cases} \end{equation}$$ { u t = Δ u − χ ∇ ⋅ (u ∇ v) + ξ ∇ ⋅ (u ∇ w) in Ω × (0 , T max) , τ v t = Δ v − φ (t , v) + f (u) in Ω × (0 , T max) , τ w = Δ w − ψ (t , w) + g (u) in Ω × (0 , T max). In this problem, Ω is a bounded and smooth domain of $ \mathbb R^n $ R n , for $ n\geq 2 $ n ≥ 2 , $ \chi,\xi \gt 0 $ χ , ξ > 0 , $ f(u) $ f (u) , $ g(u) $ g (u) reasonably regular functions generalizing, respectively, the prototypes $ f(u)=\alpha u^k $ f (u) = α u k and $ g(u)= \gamma u^l $ g (u) = γ u l , for some $ k,l,\alpha,\gamma \gt 0 $ k , l , α , γ > 0 and all $ u\geq 0 $ u ≥ 0. Moreover, $ \varphi (t,v) $ φ (t , v) and $ \psi (t,w) $ ψ (t , w) have specific expressions, $ \tau \in \{0,1\} $ τ ∈ { 0 , 1 } and $ \Theta :=\chi \alpha -\xi \gamma $ Θ := χ α − ξ γ. Once for any sufficiently smooth $ u(x,0)=u_0(x)\geq 0 $ u (x , 0) = u 0 (x) ≥ 0 , $ \tau v(x,0)=\tau v_0(x)\geq 0 $ τ v (x , 0) = τ v 0 (x) ≥ 0 and $ \tau w(x,0)=\tau w_0(x)\geq 0, $ τ w (x , 0) = τ w 0 (x) ≥ 0 , the local well-posedness of problem ( $ \Diamond $ ◊) is ensured, and we establish for the classical solution $ (u,v,w) $ (u , v , w) defined in $ \Omega \times (0,T_{\max }) $ Ω × (0 , T max) that the life span is indeed $ T_{\max }=\infty $ T max = ∞ and u, v and w are uniformly bounded in $ \Omega \times (0,\infty) $ Ω × (0 , ∞) in the following cases: For $ \varphi (t,v)=\beta v $ φ (t , v) = β v , $ \beta \gt 0 $ β > 0 , $ \psi (t,w)=\delta w $ ψ (t , w) = δ w , $ \delta \gt 0 $ δ > 0 and $ \tau =0 $ τ = 0 , provided (I.1) k 0 , $ \psi (t,w)=\delta w $ ψ (t , w) = δ w , $ \delta \gt 0 $ δ > 0 and $ \tau =1 $ τ = 1 , whenever (II.1) $ l,k\in \left (0,\frac {1}{n}\right ] $ l , k ∈ (0 , 1 n ] ; (II.2) $ l\in \left (\frac {1}{n},\frac {1}{n}+\frac {2}{n^2+4}\right) $ l ∈ (1 n , 1 n + 2 n 2 + 4) and $ k\in \left (0,\frac {1}{n}\right ] $ k ∈ (0 , 1 n ] , or $ k\in \left (\frac {1}{n},\frac {1}{n}+\frac {2}{n^2+4}\right) $ k ∈ (1 n , 1 n + 2 n 2 + 4) and $ l\in \left (0,\frac {1}{n}\right ] $ l ∈ (0 , 1 n ] ; (II.3) $ l,k\in \left (\frac {1}{n},\frac {1}{n}+\frac {2}{n^2+4}\right) $ l , k ∈ (1 n , 1 n + 2 n 2 + 4) . For $ \varphi (t,v)=\frac {1}{|\Omega |}\int _\Omega f(u) $ φ (t , v) = 1 | Ω | ∫ Ω f (u) and $ \psi (t,w)=\frac {1}{|\Omega |}\int _\Omega g(u) $ ψ (t , w) = 1 | Ω | ∫ Ω g (u) and $ \tau =0 $ τ = 0 , under the assumptions k

In this paper, we consider the Cauchy problem of the 3D incompressible Keller–Segel–Navier–Stokes equations with partial diffusion, namely we remove the diffusion $ \Delta \rho $ Δ ρ. Using the damping effect of the growth term $ -\rho ^{3} $ − ρ 3 and the geometry of axisymmetric flow without swirl, we prove the global existence of weak solutions for the system. [ABSTRACT FROM AUTHOR]

In this paper, we study blowup properties of weak solutions in their W 1 , ∞ norm to the degenerate parabolic equation with multi-nonlinearities and gradient terms. First, we show the existence and uniqueness of weak solutions by using the priori estimate methods. Second, we obtain the global existence criteria and blowup criteria after proving some gradient estimates for different coefficients. Third, we use some Sobolev's inequalities and deal with some differential inequalities of new barrier functions to determine some upper and lower bounds of blowup time of solutions. It could be found out that the blowup or global existence phenomena depend sensitively on the relationship among the different exponents of nonlinearities, which discover a key clue to the different effect of diffusion, gradient term, source term and absorption term on the singular properties of weak solutions. [ABSTRACT FROM AUTHOR]

This paper studies a fully parabolic chemotaxis system with competitive kinetics and fractional diffusion of order α 1 , α 2 ∈ (0 , 2) { u t = − d 1 Λ α 1 u − χ 1 ∇ ⋅ (u ∇ w) + μ 1 u (1 − u − a 1 v) , x ∈ T 2 , t > 0 , v t = − d 2 Λ α 2 v − χ 2 ∇ ⋅ (v ∇ w) + μ 2 v (1 − v − a 2 u) , x ∈ T 2 , t > 0 , w t = d 3 Δ w − γ 1 w + γ 2 u + γ 3 v , x ∈ T 2 , on two dimensional periodic torus T 2 . It is proved that (1) has a unique global bounded solution for all appropriately regular nonnegative initial data u 0 , v 0 and w 0 . Moreover, if μ 1 and μ 2 ( μ 1 or μ 2 ) are large enough, we can reach the following conclusions by constructing appropriate energy functional: (i) 0 < a 1 , a 2 < 1 (u , v , w) (⋅ , t) → (1 − a 1 1 − a 1 a 2 , 1 − a 2 1 − a 1 a 2 , γ 2 (1 − a 1) + γ 2 (1 − a 2) γ 1 (1 − a 1 a 2)) uniformly in T 2 as t → ∞. (ii) a 1 ≥ 1 , 0 < a 2 < 1 (u , v , w) (⋅ , t) → (0 , 1 , γ 3 γ 1 ) uniformly in T 2 as t → ∞. (iii) 0 < a 1 < 1 , a 2 ≥ 1 (u , v , w) (⋅ , t) → (1 , 0 , γ 2 γ 1 ) uniformly in T 2 as t → ∞. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study the Cauchy problem of a d-dimensional Navier–Stokes–Nernst–Planck–Poisson equation in Fourier–Besov space. Based on its special structure, the assumption of local smallness of the initial data can be ignored to obtain the global well-posedness, and it is proved that the global existence of the solution can be obtained only if part of the initial data is small enough. [ABSTRACT FROM AUTHOR]

In this paper, we will study local and global Cauchy problems for the semilinear parabolic equations ∂ t u − Δ u = G (u) with initial data in Herz spaces. These spaces unify and generalize many classical function spaces such as Lebesgue spaces of power weights. Our results cover the results obtained with initial data in Lebesgue spaces. Moreover, the results in Herz spaces are a little different from the results in Lebesgue spaces. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of the initial-nonlinear nonlocal solutions for a parabolic system in a weighted Sobolev space. The methods applied are the Faedo–Galerkin approximation and the general theory of weak compactness in appropriate weighted Sobolev spaces together with using the Poincaré-type operator for dealing nonlinear nonlocal conditions. Furthermore, the boundedness and positivity of solutions depending on the boundedness and positivity of given data are also discussed by using suitable test functions. [ABSTRACT FROM AUTHOR]

This paper deals with the parabolic equations coupled via nonlocal multiple nonlinearity. We determine the critical Fujita exponents for blow-up solutions, which are influenced by the time coefficients. Moreover, we obtain the complete and optimal classification on simultaneous and non-simultaneous blow-up of solutions. Then simultaneous blow-up rates and estimates on blow-up time are studied according to the precise blow-up phenomena for all of the dimensions of the space domain. [ABSTRACT FROM AUTHOR]

Recently, in [Tang GJ, Cen JX, Nguyen VT, et al. Differential variational-hemivariational inequalities: existence, uniqueness, stability, and convergence. J Fixed Point Appl. 2020; DOI: ],we studied a comprehensive system called differential variational-hemivar-iational inequality (DVHVI, for short) which is composed of a nonlinear evolution equation and a time-dependent variational-hemivariational inequality in Banach spaces. We have proved the existence, uniqueness, and stability of the solution in mild sense, as well as a surprising convergence result for DVHVI. However, to illustrate the applicability of those theoretical results in Tang et al., the present paper is devoted to explore a coupled dynamic system which is formulated by a nonlinear reaction–diffusion equation described by a time-dependent nonsmooth semipermeability problem. [ABSTRACT FROM AUTHOR]

In this paper, we consider a non-Newtonian fluid and electrorheological fluid mixed-type equation v t = div ⁡ (b (x) | v | α (x) | ∇ v | p − 2 ∇ v) + f (x , t , v) , where p>1, b (x) ≥ 0 and α (x) ≥ 0. The existence of weak solution is investigated by means of the parabolically regularized method. Due to | ∇ v | ∈ L ∞ (0 , T ; L l o c p (Ω)) , the stability of weak solution is established by choosing a suitable test function and the local stability is also discussed without any boundary value condition. [ABSTRACT FROM AUTHOR]

In this paper, we study some qualitative properties for solutions to an evolution problem that combines local and nonlocal diffusion operators acting in two different subdomains. The coupling takes place at the interface between these two domains in such a way that the resulting evolution problem is the gradient flow of an energy functional. We prove existence and uniqueness results, as well as that the model preserves the total mass of the initial condition. We also study the asymptotic behavior of the solutions. Besides, we show a suitable way to recover the heat equation at the whole domain from taking the limit at the nonlocal rescaled kernel. Finally, we propose a brief discussion about the extension of the problem to higher dimensions. [ABSTRACT FROM AUTHOR]

This paper is devoted to study the dynamical behavior of a Belousov–Zhabotinsky reaction–diffusion system with nonlocal effect and find the essential difference between it and classical equations. First, we prove the existence of the solution by using the comparison principle and constructing monotonic sequences. Furthermore, the uniqueness is given by using fundamental solution and Gronwall's inequality. Then we obtain the uniform boundedness of the solution by means of auxiliary function. Finally, we investigate the states of the solution as the parameters change and show some new and interesting phenomena about nonlocal Belousov–Zhabotinsky reaction–diffusion system by using stability analysis and numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, we establish blow-up rates for a higher-order semilinear parabolic equation with nonlocal in time nonlinearity with no positive assumption on the solution. We also give Liouville-type theorem for higher-order semilinear parabolic equation with infinite memory nonlinear term which plays the main tools to prove our blow-up rate result. Finally, we study the well-posedness of mild solutions. [ABSTRACT FROM AUTHOR]

This paper deals with the initial-boundary value problem for the two-species chemotaxis-competition system with two signals ∂ t u 1 = Δ u 1 − χ 11 ∇ ⋅ (u 1 ∇ v 1) − χ 12 ∇ ⋅ (u 1 ∇ v 2) + μ 1 u 1 (1 − u 1 − a 1 u 2) , ∂ t u 2 = Δ u 2 − χ 21 ∇ ⋅ (u 2 ∇ v 1) − χ 22 ∇ ⋅ (u 2 ∇ v 2) + μ 2 u 2 (1 − u 2 − a 2 u 1) , 0 = Δ v 1 − λ 1 v 1 + α 11 u 1 + α 12 u 2 , 0 = Δ v 2 − λ 2 v 2 + α 21 u 1 + α 22 u 2 , under the homogeneous Neumann boundary condition, where x ∈ Ω , t > 0 , χ i j > 0 , μ i > 0 , a i > 0 , α i j > 0 , λ i > 0 (i , j = 1 , 2) , and Ω ⊂ R n (n ≥ 2) is a smooth bounded domain. If χ 11 / μ 1 , χ 12 / μ 1 , χ 21 / μ 2 and χ 22 / μ 2 are sufficiently small, then the system possesses a globally bounded classical solution for any suitably regular initial data u 10 , u 20 . Furthermore, by constructing some appropriate functionals, it is shown that For the weak competition case, if μ 1 , μ 2 are sufficiently large, then the solution (u 1 , u 2 , v 1 , v 2) converges to 1 − a 1 1 − a 1 a 2 , 1 − a 2 1 − a 1 a 2 , α 11 (1 − a 1) + α 12 (1 − a 2) λ 1 (1 − a 1 a 2) , α 21 (1 − a 1) + α 22 (1 − a 2) λ 2 (1 − a 1 a 2) exponentially as t → ∞. For the strong-weak competition case, if μ 2 is sufficiently large, then the solution (u 1 , u 2 , v 1 , v 2) converges to (0 , 1 , α 12 / λ 1 , α 22 / λ 2) with exponential decay when a 1 > 1 , and with algebraic decay when a 1 = 1. [ABSTRACT FROM AUTHOR]

In this paper, the two-dimensional incompressible chemotaxis fluid with logical source is studied as following: n t + u ⋅ ∇ n = Δ n − ∇ ⋅ (n ∇ c) + n − n 2 , c t + u ⋅ ∇ c = Δ c − n c , u t + u ⋅ ∇ u + ∇ P = Δ u − n ∇ φ , ∇ ⋅ u = 0. By taking advantage of a coupling structure of the equations and using a scale decomposition technique, the global existence and uniqueness of weak solutions to the above system for a large class of initial data is obtained. [ABSTRACT FROM AUTHOR]

In this paper, we study the long-time behavior of the chemotaxis-Navier–Stokes system ∂ t n + u ⋅ ∇ n = λ Δ n − ∇ ⋅ (χ (c) n ∇ c) , ∂ t c + u ⋅ ∇ c = μ Δ c − f (c) n , ∂ t u + u ⋅ ∇ u + ∇ P = ζ Δ u − n ∇ φ , ∇ ⋅ u = 0 , t > 0 , x ∈ Ω posed in a strip domain Ω := R 2 × [ 0 , 1 ] ⊂ R 3 . In Peng-Xiang (Math. Models Methods Appl. Sci., 28 (2018), 869-920), the authors have established the global existence of strong solutions to this system with non-flux boundary conditions for n and c and non-slip boundary conditions for u. Our main purpose is to establish the time-decay rates for such solutions. This will be done by using the anisotropic L p interpolation and the iterative techniques. [ABSTRACT FROM AUTHOR]

In this paper, we consider a liquid–gas system with two components: water and hydrogen flow model in heterogeneous porous media with periodic microstructure taking into account kinetics in the mass transfer between the two phases. The particular feature in this model is that chemistry effects are taken into account. The microscopic model consists of the usual equations derived from the mass conservation laws of both fluids, along with the Darcy–Muskat and capillary pressure laws and the mass exchange is modeled as a source term in the equations. The problem is written in the terms of the phase formulation; i.e. the saturation of one phase, the pressure of the second phase, and the concentration of dissolved hydrogen in the liquid phase. The mathematical model consists in a system of partial differential equations: two degenerate nonlinear parabolic equations and one diffusion–convection equation. The major difficulties related to this new model are in the nonlinear degenerate structure of the equations, as well as in the coupling in the system. Under some realistic assumptions on the data, we obtain a nonlinear homogenized coupled system of three coupled partial differential equations with effective coefficients which are computed via solving cell problems. We give a rigorous mathematical derivation of the effective model by means of the two-scale convergence. [ABSTRACT FROM AUTHOR]

In this paper, we prove the global well-posedness of weak and strong solutions to the n D (n ≥ 3) phase-lock system under the Coulomb gauge. [ABSTRACT FROM AUTHOR]

In this paper, we study the evolution p, q-Laplacian equations u t = d i v (| ∇ u | p − 2 ∇ u) + u α ∫ Ω v m d x and v t = d i v (| ∇ v | q − 2 ∇ v) + v β ∫ Ω u n d x with 1 (p − 1 − α) (q − 1 − β) , there exist suitable initial data such that vanishing solutions exist. If m n < (p − 1 − α) (q − 1 − β) , we find the explicit scopes of initial data such that the solutions could not vanish, which complete the corresponding classifications of solutions in Math. Methods Appl. Sci. 39 (2016) 1325–1335 and Appl. Math. Comp. 259 (2015) 587–595, respectively. For the critical case m n = (p − 1 − α) (q − 1 − β) , the solutions vanish in finite time with small initial data. [ABSTRACT FROM AUTHOR]

This paper deals with the following two-species chemotaxis system u t = ∇ ⋅ (D 1 (u) ∇ u) − ∇ ⋅ (S 1 (u) ∇ w) + μ 1 u (1 − u − a 1 v) , x ∈ Ω , t > 0 , v t = ∇ ⋅ (D 2 (v) ∇ v) − ∇ ⋅ (S 2 (v) ∇ w) + μ 2 v (1 − a 2 u − v) , x ∈ Ω , t > 0 , w t = Δ w − (u + v) w , x ∈ Ω , t > 0 with homogeneous Neumann boundary conditions and suitable initial conditions, where Ω ⊂ R n ( n ≥ 1) is a bounded and smooth domain. Moreover, the parameters μ 1 , μ 2 > 0 , and a 1 , a 2 > 0. For i = 1, 2, we assume that D i (s) ≥ c D i (s + 1) m i − 1 a n d S i (s) ≤ c S i (s + 1) q i − 1 w i t h S i (s) ≥ 0 f o r a l l s ≥ 0 , where c D i > 0 , c S i > 0 , m i ∈ R and q i ∈ R . Then if one of the following cases holds: q i < m i 2 + 2 − n n + 2 ; q i < m i 2 + 3 2 and μ i is sufficiently large, it is proved that the system possesses a unique global bounded classical solution. [ABSTRACT FROM AUTHOR]

In this paper, a new analysis for the existence, uniqueness, and regularity of solutions to a time-dependent Kohn–Sham equation is presented. The Kohn–Sham equation is a nonlinear integral Schrödinger equation that is of great importance in many applications in physics and computational chemistry. To deal with the time-dependent, nonlinear and non-local potentials of the Kohn–Sham equation, the analysis presented in this manuscript makes use of energy estimates, fixed-point arguments, regularization techniques, and direct estimates of the non-local potential terms. The assumptions considered for the time-dependent and nonlinear potentials make the obtained theoretical results suitable to be used also in an optimal control framework. [ABSTRACT FROM AUTHOR]

This paper proves global existence and uniqueness of axially symmetric weak solutions to the 3D time-dependent Ginzburg–Landau model in superconductivity in R 3 with L 2 initial data and the choice of Lorentz gauge. We are unable to prove a similar result with the Coulomb gauge. [ABSTRACT FROM AUTHOR]