We study the behaviour, when p → + ∞ p\to +\infty , of the first p-Laplacian eigenvalues with Robin boundary conditions and the limit of the associated eigenfunctions. We prove that the limit of the eigenfunctions is a viscosity solution to an eigenvalue problem for the so-called ∞ \infty -Laplacian. Moreover, in the second part of the article, we focus our attention on the p-Poisson equation when

The article is concerned with systems of fractional discrete equations Δ α x ( n + 1 ) = F n ( n , x ( n ) , x ( n − 1 ) , … , x ( n 0 ) ) , n = n 0 , n 0 + 1 , … , {\Delta }^{\alpha }x\left(n+1)={F}_{n}\left(n,x\left(n),x\left(n-1),\ldots ,x\left({n}_{0})),\hspace{1em}n={n}_{0},{n}_{0}+1,\ldots , where n 0 ∈ Z {n}_{0}\in {\mathbb{Z}} , n n is an independent variable, Δ α {\Delta }^{\alpha } is an

The existence of unbounded solutions and their asymptotic behavior is studied for higher order differential equations considered as perturbations of certain linear differential equations. In particular, the existence of solutions with polynomial-like or noninteger power-law asymptotic behavior is proved. These results give a relation between solutions to nonlinear and corresponding linear equations

This paper is devoted to investigate the existence and multiplicity of the normalized solutions for the following fractional Schrödinger equation: (P) ( − Δ ) s u + λ u = μ ∣ u ∣ p − 2 u + ∣ u ∣ 2 s ∗ − 2 u , x ∈ R N , u > 0 , ∫ R N ∣ u ∣ 2 d x = a 2 , \left\{\begin{array}{l}{\left(-\Delta )}^{s}u+\lambda u=\mu | u{| }^{p-2}u+| u{| }^{{2}_{s}^{\ast }-2}u,\hspace{1em}x\in {{\mathbb{R}}}^{N},\hspace{1

Let n ≥ 2 n\ge 2 and Ω ⊂ R n \Omega \subset {{\mathbb{R}}}^{n} be a bounded nontangentially accessible domain. In this article, the authors investigate (weighted) global gradient estimates for Dirichlet boundary value problems of second-order elliptic equations of divergence form with an elliptic symmetric part and a BMO antisymmetric part in Ω \Omega . More precisely, for any given p ∈ ( 2 , ∞ ) p\in

In this article, we consider nondiffusive variational problems with mixed boundary conditions and (distributional and weak) gradient constraints. The upper bound in the constraint is either a function or a Borel measure, leading to the state space being a Sobolev one or the space of functions of bounded variation. We address existence and uniqueness of the model under low regularity assumptions, and

In this work, we handle a time-dependent Navier-Stokes problem in dimension three with a mixed boundary conditions. The variational formulation is written considering three independent unknowns: vorticity, velocity, and pressure. We use the backward Euler scheme for time discretization and the spectral method for space discretization. We present a complete numerical analysis linked to this variational

This article is concerned with the following nonlinear supercritical elliptic problem: − M ( ‖ ∇ u ‖ 2 2 ) Δ u = f ( x , u ) , in B 1 ( 0 ) , u = 0 , on ∂ B 1 ( 0 ) , \left\{\begin{array}{ll}-M(\Vert \nabla u{\Vert }_{2}^{2})\Delta u=f\left(x,u),& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{B}_{1}\left(0),\\ u=0,& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial {B}_{1}\left(0)

We prove long-time existence for the negative L 2 {L}^{2} -gradient flow of the p-elastic energy, p ≥ 2 p\ge 2 , with an additive positive multiple of the length of the curve. To achieve this result, we regularize the energy by cutting off the degeneracy at points with vanishing curvature and add a small multiple of a higher order energy, namely, the square of the L 2 {L}^{2} -norm of the normal gradient

We study positive solutions to the one-dimensional generalized double phase problems of the form: − ( a ( t ) φ p ( u ′ ) + b ( t ) φ q ( u ′ ) ) ′ = λ h ( t ) f ( u ) , t ∈ ( 0 , 1 ) , u ( 0 ) = 0 = u ( 1 ) , \left\{\begin{array}{l}-(a\left(t){\varphi }_{p}\left(u^{\prime} )+b\left(t){\varphi }_{q}\left(u^{\prime} ))^{\prime} =\lambda h\left(t)f\left(u),\hspace{1em}t\in \left(0,1),\\ u\left(0)=0=u\left(1)

We study robust regularity estimates for a class of nonlinear integro-differential operators with anisotropic and singular kernels. In this paper, we prove a Sobolev-type inequality, a weak Harnack inequality, and a local Hölder estimate.

In this article, we study a large class of evolutionary variational–hemivariational inequalities of hyperbolic type without damping terms, in which the functional framework is considered in an evolution triple of spaces. The inequalities contain both a convex potential and a locally Lipschitz superpotential. The results on existence, uniqueness, and regularity of solution to the inequality problem

We consider Hamiltonian functions of the classical type, namely, even and convex with respect to the generalized momenta. A brake orbit is a periodic solution of Hamilton’s equations such that the generalized momenta are zero on two different points. Under mild assumptions, this paper reduces the multiplicity problem of the brake orbits for a Hamiltonian function of the classical type to the multiplicity

In this article, we consider a class of optimal control problems governed by state equations of Kobayashi-Warren-Carter-type. The control is given by physical temperature. The focus is on problems in dimensions less than or equal to 4. The results are divided into four Main Theorems, concerned with: solvability and parameter dependence of state equations and optimal control problems; the first-order

In this article, we are interested in the following Kirchhoff-type problem (0.1) − a + b ∫ R N ∣ ∇ u ∣ 2 d x Δ u + V ( ∣ x ∣ ) u = ∣ u ∣ 2 u in R N , u ∈ H 1 ( R N ) , \left\{\begin{array}{l}-\left(a+b\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}| \nabla u\hspace{-0.25em}{| }^{2}{\rm{d}}x\right)\Delta u+V\left(| x| )u=| u\hspace{-0.25em}{| }^{2}u\hspace{1.0em}{\rm{in}}\hspace{0.33em}{{\mathbb{R}}}^{N}

In this article, we consider the upper critical Choquard equation with a local perturbation − Δ u = λ u + ( I α ∗ ∣ u ∣ p ) ∣ u ∣ p − 2 u + μ ∣ u ∣ q − 2 u , x ∈ R N , u ∈ H 1 ( R N ) , ∫ R N ∣ u ∣ 2 = a , \left\{\begin{array}{l}-\Delta u=\lambda u+\left({I}_{\alpha }\ast | u\hspace{-0.25em}{| }^{p})| u\hspace{-0.25em}{| }^{p-2}u+\mu | u\hspace{-0.25em}{| }^{q-2}u,\hspace{1em}x\in {{\mathbb{R}}}^{N}

In this article, we consider the parabolic system ( u i ) t = ∇ ⋅ ( m U m − 1 A ( ∇ u i , u i , x , t ) + ℬ ( u i , x , t ) ) , ( 1 ≤ i ≤ k ) {({u}^{i})}_{t}=\nabla \cdot (m{U}^{m-1}{\mathcal{A}}(\nabla {u}^{i},{u}^{i},x,t)+{\mathcal{ {\mathcal B} }}({u}^{i},x,t)),\hspace{1.0em}(1\le i\le k) in the range of exponents m > n − 2 n m\gt \frac{n-2}{n} where the diffusion coefficient U U depends on the

We consider the Dirichlet problem for partial trace operators which include the smallest and the largest eigenvalue of the Hessian matrix. It is related to two-player zero-sum differential games. No Lipschitz regularity result is known for the solutions, to our knowledge. If some eigenvalue is missing, such operators are nonlinear, degenerate, non-uniformly elliptic, neither convex nor concave. Here

In this article, a new type of comparison theorem for some second-order nonlinear parabolic systems with nonlinear boundary conditions is given, which can cover classical linear boundary conditions, such as the homogeneous Dirichlet or Neumann boundary condition. The advantage of our comparison theorem over the classical ones lies in the fact that it enables us to compare two solutions satisfying different

In this article, we consider the non-linear Choquard equation − Δ u + V ( ∣ x ∣ ) u = ∫ R 3 ∣ u ( y ) ∣ 2 ∣ x − y ∣ d y u in R 3 , -\Delta u+V\left(| x| )u=\left(\mathop{\int }\limits_{{{\mathbb{R}}}^{3}}\frac{| u(y){| }^{2}}{| x-y| }{\rm{d}}y\right)u\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3}, where V ( r ) V\left(r) is a positive bounded function. Under some

In this article, we consider the following Schrödinger-Poisson system: − Δ u + u + k ( x ) ϕ ( x ) u = f ( x ) ∣ u ∣ p − 1 u + g ( x ) , x ∈ R 3 , − Δ ϕ = k ( x ) u 2 , x ∈ R 3 , \left\{\begin{array}{ll}-\Delta u+u+k\left(x)\phi \left(x)u=f\left(x)| u{| }^{p-1}u+g\left(x),& x\in {{\mathbb{R}}}^{3},\\ -\Delta \phi =k\left(x){u}^{2},& x\in {{\mathbb{R}}}^{3},\end{array}\right. with p ∈ ( 3 , 5 ) p\in

This article deals with the following fractional Kirchhoff problem with critical exponent a + b ∫ R N ∣ ( − Δ ) s 2 u ∣ 2 d x ( − Δ ) s u = ( 1 + ε K ( x ) ) u 2 s ∗ − 1 , in R N , \left(a+b\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}| {\left(-\Delta )}^{\tfrac{s}{2}}u\hspace{-0.25em}{| }^{2}{\rm{d}}x\right){\left(-\Delta )}^{s}u=\left(1+\varepsilon K\left(x)){u}^{{2}_{s}^{\ast }-1},\hspace{1.0em}\hspace{0

In this paper we are concerned with the existence, nonexistence and bifurcation of nontrivial solution of the nonlinear Schrödinger-Korteweg-de Vries type system(NLS-NLS-KdV). First, we find some conditions to guarantee the existence and nonexistence of positive solution of the system. Second, we study the asymptotic behavior of the positive ground state solution. Finally, we use the classical Crandall-Rabinowitz

We first prove De Giorgi type level estimates for functions in W 1, t (Ω), Ω⊂RN $ \Omega\subset{\mathbb R}^N $ , with t>N≥2 $ t \gt N\geq 2 $ . This augmented integrability enables us to establish a new Harnack type inequality for functions which do not necessarily belong to De Giorgi’s classes as obtained in Di Benedetto–Trudinger [10] for functions in W 1,2 (Ω). As a consequence, we prove the validity

Abstract In this article, we aimed to study a class of nonhomogeneous fractional (p, q)-Laplacian systems with critical nonlinearities as well as critical Hardy nonlinearities in R N {{\mathbb{R}}}^{N} . By appealing to a fixed point result and fractional Hardy-Sobolev inequality, the existence of nontrivial nonnegative solutions is obtained. In particular, we also consider Choquard-type nonlinearities

Abstract In this article, we study discrete Kirchhoff-type problems when the nonlinearity is resonant at both zero and infinity. We establish a series of results on the existence of nontrivial solutions by combining variational method with Morse theory. Several examples are provided to illustrate applications of our results.

Abstract Using elements of the theory of linear operators in Hilbert spaces and monotonicity tools we obtain the existence and uniqueness results for a wide class of nonlinear problems driven by the equation T x = N ( x ) Tx=N\left(x) , where T T is a self-adjoint operator in a real Hilbert space ℋ {\mathcal{ {\mathcal H} }} and N N is a nonlinear perturbation. Both potential and nonpotential perturbations

Abstract This article deals with the degenerate fractional Kirchhoff wave equation with structural damping or strong damping. The well-posedness and the existence of global attractor in the natural energy space by virtue of the Faedo-Galerkin method and energy estimates are proved. It is worth mentioning that the results of this article cover the case of possible degeneration (or even negativity) of

In the present paper, a class of Schrödinger equations is investigated, which can be stated as −Δu+V(x)u=f(u), x∈ℝN. - \Delta u + V(x)u = f(u),\;\;\;\;x \in {{\rm{\mathbb R}}^N}. If the external potential V is radial and coercive, then we give the local Ambrosetti-Rabinowitz super-linear condition on the nonlinearity term f ∈ C (ℝ, ℝ) which assures the problem has not only infinitely many radial sign-changing

We consider on M (ℝ d ) (the set of all finite measures on ℝ d ) the evolution equation associated with the nonlinear operator F↦ΔF′+∑k⩾1bkFk F \mapsto \Delta F' + \sum\nolimits_{k \geqslant 1} b_k F^k , where F′ is the variational derivative of F and we show that it has a solution represented by means of the distribution of the d -dimensional Brownian motion and the non-local branching process on

We consider density solutions for gradient flow equations of the form u t = ∇ · ( γ ( u )∇ N( u )), where N is the Newtonian repulsive potential in the whole space ℝ d with the nonlinear convex mobility γ ( u ) = u α , and α > 1. We show that solutions corresponding to compactly supported initial data remain compactly supported for all times leading to moving free boundaries as in the linear mobility

In this paper we study the existence and the nonexistence of solutions to an inhomogeneous non-linear elliptic problem (P) −Δu+u=F(u)+κμ in RN, u>0 in RN, u(x)→0 as |x|→∞, - \Delta u + u = F(u) + \kappa \mu \quad {\kern 1pt} {\rm in}{\kern 1pt} \quad {{\bf R}^N},\quad u > 0\quad {\kern 1pt} {\rm in}{\kern 1pt} \quad {{\bf R}^N},\quad u(x) \to 0\quad {\kern 1pt} {\rm as}{\kern 1pt} \quad |x| \to \infty

Let X be the direct product of X i where X i is smooth manifold for 1 ≤ i ≤ k . As is known, if every X i satisfies the doubling volume condition, then the centered Hardy-Littlewood maximal function M on X is weak (1,1) bounded. In this paper, we consider the product manifold X where at least one X i does not satisfy the doubling volume condition. To be precise, we first investigate the mapping properties

Asymptotic properties of the sequences ( a ) {Pj}j=1∞ $\{P^{j}\}_{j=1}^{\infty}$and ( b ) {j−1∑i=0j−1Pi}j=1∞ $\{ j^{-1} \sum _{i=0}^{j-1} P^{i}\}_{j=1}^{\infty}$ are studied for g ∈ G = { f ∈ L 1 ( I ) : f ≥ 0 and ‖ f ‖ = 1}, where P : L 1 ( I ) → L 1 ( I ) is a Markov operator defined by Pf:=∫Pyfdp(y) $Pf:= \int P_{y}f\, dp(y) $for f ∈ L 1 ; { P y } y∈Y is the family of the Frobenius-Perron operators

For nonautonomous linear difference equations, we introduce the notion of the so-called nonuniform dichotomy spectrum and prove a spectral theorem. As an application of the spectral theorem, we prove a reducibility result.

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

Let w be a Muckenhoupt A 2 (ℝ n ) weight and Ω a bounded Reifenberg flat domain in ℝ n . Assume that p (·):Ω → (1, ∞) is a variable exponent satisfying the log-Hölder continuous condition. In this article, the authors investigate the weighted W 1, p (·) (Ω, w )-regularity of the weak solutions of second order degenerate elliptic equations in divergence form with Dirichlet boundary condition, under

In the paper, nonlinear systems of mixed-type functional differential equations are analyzed and the existence of semi-global and global solutions is proved. In proofs, the monotone iterative technique and Schauder-Tychonov fixed-point theorem are used. In addition to proving the existence of global solutions, estimates of their co-ordinates are derived as well. Linear variants of results are considered

We study distributional solutions of semilinear biharmonic equations of the type Δ2u+f(u)=0 onℝN, {\Delta ^2}u + f(u) = 0\quad on\;{{\mathbb R} ^N}, where f is a continuous function satisfying f ( t ) t ≥ c | t | q +1 for all t ∈ ℝ with c > 0 and q > 1. By using a new approach mainly based on careful choice of suitable weighted test functions and a new version of Hardy- Rellich inequalities, we prove

This paper provides the Carleson characterization of the extension of fractional Sobolev spaces and Lebesgue spaces to Lq(ℝ+n+1,μ) L^q (\mathbb{R}_ + ^{n + 1} ,\mu ) via space-time fractional equations. For the extension of fractional Sobolev spaces, preliminary results including estimates, involving the fractional capacity, measures, the non-tangential maximal function, and an estimate of the Riesz

We show existence and concentration results for a class of p & q critical problems given by −divaϵp|∇u|pϵp|∇u|p−2∇u+V(z)b|u|p|u|p−2u=f(u)+|u|q⋆−2uinRN, $$-div\left(a\left(\epsilon^{p}|\nabla u|^{p}\right) \epsilon^{p}|\nabla u|^{p-2} \nabla u\right)+V(z) b\left(|u|^{p}\right)|u|^{p-2} u=f(u)+|u|^{q^{\star}-2} u\, \text{in} \,\mathbb{R}^{N},$$ where u ∈ W 1, p (ℝ N ) ∩ W 1, q (ℝ N ), ϵ > 0 is a small

In this paper we provide a numerical approximation of bifurcation branches from nodal radial solutions of the Lane Emden Dirichlet problem in the unit ball in ℝ 2 , as the exponent of the nonlinearity varies. We consider solutions with two or three nodal regions. In the first case our numerical results complement the analytical ones recently obtained in [11]. In the case of solutions with three nodal

We give a new non-smooth Clark’s theorem without the global symmetric condition. The theorem can be applied to generalized quasi-linear elliptic equations with small continous perturbations. Our results improve the abstract results about a semi-linear elliptic equation in Kajikiya [10] and Li-Liu [11].

In this paper we study double phase problems with nonlinear boundary condition and gradient dependence. Under quite general assumptions we prove existence results for such problems where the perturbations satisfy a suitable behavior in the origin and at infinity. Our proofs make use of variational tools, truncation techniques and comparison methods. The obtained solutions depend on the first eigenvalues

In this paper, we establish the second boundary behavior of the unique strictly convex solution to a singular Dirichlet problem for the Monge-Ampère equation det(D2u)=b(x)g(−u),u<0 in Ω and u=0 on ∂Ω, $$\mbox{ det}(D^{2} u)=b(x)g(-u),\,u<0 \mbox{ in }\Omega \mbox{ and } u=0 \mbox{ on }\partial\Omega, $$ where Ω is a bounded, smooth and strictly convex domain in ℝ N ( N ≥ 2), b ∈ C ∞ (Ω) is positive

We consider a nonlinear boundary value problem with degenerate nonlocal term depending on the L q -norm of the solution and the p -Laplace operator. We prove the multiplicity of positive solutions for the problem, where the number of solutions doubles the number of “positive bumps” of the degenerate term. The solutions are also ordered according to their L q -norms.

In the present paperwe study the existence of nontrivial solutions of a class of static coupled nonlinear fractional Hartree type system. First, we use the direct moving plane methods to establish the maximum principle(Decay at infinity and Narrow region principle) and prove the symmetry and nonexistence of positive solution of this nonlocal system. Second, we make complete classification of positive

We are concerned with the following Schrödinger system with coupled quadratic nonlinearity −ε2Δv+P(x)v=μvw,x∈RN,−ε2Δw+Q(x)w=μ2v2+γw2,x∈RN,v>0,w>0,v,w∈H1RN, $$\begin{equation}\left\{\begin{array}{ll}-\varepsilon^{2} \Delta v+P(x) v=\mu v w, & x \in \mathbb{R}^{N}, \\ -\varepsilon^{2} \Delta w+Q(x) w=\frac{\mu}{2} v^{2}+\gamma w^{2}, & x \in \mathbb{R}^{N}, \\ v>0, \quad w>0, & v, w \in H^{1}\left(\

In this work, we study the existence of a positive solution to an elliptic equation involving the fractional Laplacian (− Δ ) s in ℝ n , for n ≥ 2, such as (0.1) (−Δ)su+E(x)u+V(x)uq−1=K(x)f(u)+u2s⋆−1. $$(-\Delta)^{s} u+E(x) u+V(x) u^{q-1}=K(x) f(u)+u^{2_{s}^{\star}-1}.$$ Here, s ∈ (0, 1), q∈2,2s⋆ $q \in\left[2,2_{s}^{\star}\right)$with 2s⋆:=2nn−2s $2_{s}^{\star}:=\frac{2 n}{n-2 s}$being the fractional

We consider a double-phase non-Newtonian fluid, described by a stress tensor which is the sum of a p -Stokes and a q -Stokes stress tensor, with 1 < p <2 < q <∞. For a wide range of parameters ( p , q ), we prove the uniqueness of small solutions. We use the p < 2 features to obtain quadratic-type estimates for the stress-tensor, while we use the improved regularity coming from the term with q > 2

This paper is concerned with the following Hamiltonian elliptic system −Δu+V(x)u=Wv(x,u,v), x∈RN,−Δv+V(x)v=Wu(x,u,v), x∈RN, $$ \left\{ \begin{array}{ll} -\Delta u+V(x)u=W_{v}(x, u, v), \ \ \ \ x\in {\mathbb{R}}^{N},\\ -\Delta v+V(x)v=W_{u}(x, u, v), \ \ \ \ x\in {\mathbb{R}}^{N},\\ \end{array} \right. $$ where z = ( u , v ) : ℝ N → ℝ 2 , V ( x ) and W ( x , z ) are 1-periodic in x . By making use of

In this paper, we are concerned with the following a new critical nonlocal Schrödinger-Poisson system on the Heisenberg group: −a−b∫Ω|∇Hu|2dξΔHu+μϕu=λ|u|q−2u+|u|2u,inΩ,−ΔHϕ=u2,inΩ,u=ϕ=0,on∂Ω, $$\begin{equation*}\begin{cases} -\left(a-b\int_{\Omega}|\nabla_{H}u|^{2}d\xi\right)\Delta_{H}u+\mu\phi u=\lambda|u|^{q-2}u+|u|^{2}u,\quad &\mbox{in} \, \Omega,\\ -\Delta_{H}\phi=u^2,\quad &\mbox{in}\, \Omega

We are devoted to the study of a semilinear time fractional Rayleigh-Stokes problem on ℝ N , which is derived from a non-Newtonain fluid for a generalized second grade fluid with Riemann-Liouville fractional derivative. We show that a solution operator involving the Laplacian operator is very effective to discuss the proposed problem. In this paper, we are concerned with the global/local well-posedness

We study the following Kirchhoff type equation: −a+b∫RN|∇u|2dxΔu+u=k(x)|u|p−2u+m(x)|u|q−2u in RN, $$\begin{equation*}\begin{array}{ll} -\left(a+b\int\limits_{\mathbb{R}^{N}}|\nabla u|^{2}\mathrm{d}x\right)\Delta u+u =k(x)|u|^{p-2}u+m(x)|u|^{q-2}u~~\text{in}~~\mathbb{R}^{N}, \end{array} \end{equation*}$$ where N =3, a,b>0 $ a,b \gt 0 $ , 1

We consider quasilinear elliptic systems in divergence form. In general, we cannot expect that weak solutions are locally bounded because of De Giorgi’s counterexample. Here we assume that off-diagonal coefficients have a “butterfly support”: this allows us to prove local boundedness of weak solutions.

In this paper, we consider the following modified quasilinear problem: − Δ u− κ uΔ u2=λ a(x)u− α +b(x)uβ inΩ ,u> 0inΩ ,u=0on∂ Ω , $$\begin{array}{} \left\{\begin{array}{c}\, -{\it\Delta} u-\kappa u{\it\Delta} u^2 = \lambda a(x)u^{-\alpha}+b(x)u^\beta \, \, in\, {\it\Omega}, \\\!\! u \gt 0 \, \, in\, {\it\Omega}, \, \, \, \, \, \, \, u = 0 \, \, on \, \partial{\it\Omega} , \\ \end{array}\right. \end{array}

In this paper, the Cauchy problem for the one-dimensional compressible isentropic magnetohydrodynamic (MHD) equations with no vacuum at infinity is considered, but the initial vacuum can be permitted inside the region. By deriving a priori ν (resistivity coefficient)-independent estimates, we establish the non-resistive limit of the global strong solutions with large initial data. Moreover, as a by-product

The paper studies absolute stability of neutral differential nonlinear systems x˙(t)=Axt+Bxt−τ+Dx˙t−τ+bf(σ(t)),σ(t)=cTx(t),t⩾0 $$ \begin{align}\dot x(t)=Ax\left ( t \right )+Bx\left ( {t-\tau} \right ) +D\dot x\left ( {t-\tau} \right ) +bf({\sigma (t)}),\,\, \sigma (t)=c^Tx(t), \,\, t\geqslant 0 \end{align} $$ where x is an unknown vector, A , B and D are constant matrices, b and c are column constant

The existence of a positive entire weak solution to a singular quasi-linear elliptic system with convection terms is established, chiefly through perturbation techniques, fixed point arguments, and a priori estimates. Some regularity results are then employed to show that the obtained solution is actually strong.

We investigate optimal decay rates for higher–order spatial derivatives of strong solutions to the 3D Cauchy problem of the compressible viscous quantum magnetohydrodynamic model in the H 5 × H 4 × H 4 framework, and the main novelty of this work is three–fold: First, we show that fourth order spatial derivative of the solution converges to zero at the L2-rate (1+t)-114 {L^2} - {\rm{rate}}\,{(1 + t)^{-