Blow‐up for wave equation with the scale‐invariant damping and combined nonlinearities

In this article, we study the blow‐up of the damped wave equation in the scale‐invariant case and in the presence of two nonlinearities. More precisely, we consider the following equation: $$u_{tt}?\mathrm{\Delta }u+\frac{\mu }{1+t}{u}_t=|u_t{|}^p+|u{|}^q,\text{in}{?}^N\times [0,\infty ),$$ with small initial data. For $\mu <\frac{N(q?1)}{2}$ and μ ∈ (0, μ_?) , where μ_? > 0 is depending on the nonlinearties' powers and the space dimension (μ_? satisfies $(q?1)\left((N+2{\mu }_{?}?1)p?2\right)=4$), we prove that the wave equation, in this case, behaves like the one without dissipation (μ = 0 ). Our result completes the previous studies in the case where the dissipation is given by $\frac{\mu }{{(1+t)}^{\beta }}{u}_t;\beta >1$, where, contrary to what we obtain in the present work, the effect of the damping is not significant in the dynamics. Interestingly, in our case, the influence of the damping term $\frac{\mu }{1+t}{u}_t$ is important.     

A critical Kirchhoff‐type problem driven by a p (·)‐fractional Laplace operator with variable s (·) ‐order

The paper deals with the following Kirchhoff‐type problem $$\left\{\begin{array}{lll}& M\left({?}_{{?}^{2N}}\frac{1}{p(x,y)}\frac{|v(x)?v(y){|}^{p(x,y)}}{|x?y{|}^{N+p(x,y)s(x,y)}}dxdy\right){(?\mathrm{\Delta })}_{p(·)}^{s(·)}v(x)=\mu g(x,v)+|v{|}^{r(x)?2}v& \text{in}\mathrm{\Omega },\\ & v=0& \text{in}{?}^N\backslash \mathrm{\Omega },\end{array}\right.$$ where M models a Kirchhoff coefficient, ${(?\mathrm{\Delta })}_{p(·)}^{s(·)}$ is a variable s(·) ‐order p(·) ‐fractional Laplace operator, with $s(·):{?}^{2N}\to (0,1)$ and $p(·):{?}^{2N}\to (1,\infty )$. Here, $\mathrm{\Omega }?{?}^N$ is a bounded smooth domain with N > p(x, y)s(x, y) for any $(x,y)\in \overline{\mathrm{\Omega }}\times \overline{\mathrm{\Omega }}$, μ is a positive parameter, g is a continuous and subcritical function, while variable exponent r(x) could be close to the critical exponent $p_s^{?}(x)=N\overline{p}(x)/(N?\overline{s}(x)\overline{p}(x))$, given with $\overline{p}(x)=p(x,x)$ and $\overline{s}(x)=s(x,x)$ for $x\in \overline{\mathrm{\Omega }}$. We prove the existence and asymptotic behavior of at least one non‐trivial solution. For this, we exploit a suitable tricky step analysis of the critical mountain pass level, combined with a Brézis and Lieb‐type lemma for fractional Sobolev spaces with variable order and variable exponent.     

Unique weak solutions of the d‐dimensional micropolar equation with fractional dissipation

This article examines the existence and uniqueness of weak solutions to the d‐dimensional micropolar equations (d=2 or d=3) with general fractional dissipation (?Δ)^αu and (?Δ)^βw. The micropolar equations with standard Laplacian dissipation model fluids with microstructure. The generalization to include fractional dissipation allows simultaneous study of a family of equations and is relevant in some physical circumstances. We establish that, when $\alpha \ge \frac{1}{2}$ and $\beta \ge \frac{1}{2}$, any initial data (u₀,w₀) in the critical Besov space $u_0\in B_{2,1}^{1+\frac{d}{2}?2\alpha }({?}^d)$ and $w_0\in B_{2,1}^{1+\frac{d}{2}?2\beta }({?}^d)$ yields a unique weak solution. For α ≥ 1 and β=0, any initial data $u_0\in B_{2,1}^{1+\frac{d}{2}?2\alpha }({?}^d)$ and $w_0\in B_{2,1}^{\frac{d}{2}}({?}^d)$ also leads to a unique weak solution as well. The regularity indices in these Besov spaces appear to be optimal and can not be lowered in order to achieve the uniqueness. Especially, the 2D micropolar equations with the standard Laplacian dissipation, namely, α=β=1, have a unique weak solution for $(u_0,w_0)\in B_{2,1}^0$. The proof involves the construction of successive approximation sequences and extensive a priori estimates in Besov space settings.     

Existence and exponential decay for a nonlinear wave equation with nonlocal boundary conditions of 2N‐point type

This paper is devoted to the study of a nonlinear wave equation with initial conditions and nonlocal boundary conditions of 2N‐point type, which connect the values of an unknown function u(x,t) at x = 1, x = 0, x = η_i(t) , and x = θ_i(t), where $0<{\eta }_1(t)<{\eta }_2(t)<\dots <{\eta }_{{}_{N?1}}(t)<1,$ $0<{\theta }_1(t)<{\theta }_2(t)<\dots <{\theta }_{{}_{N?1}}(t)<1,$ for all t ≥ 0. First, we prove local existence of a unique weak solution by using density arguments and applying the Banach's contraction principle. Next, under the suitable conditions, we show that the problem considered has a unique global solution u(t) with energy decaying exponentially as t → +∞ . Finally, we present numerical results.     

Norm of a multilinear integral operator from product of weighted-type spaces to weighted-type space

We investigate the following multilinear integral operator $$T_K^m(\overrightarrow{f})(x)={\int }_0^{\infty }\dots {\int }_0^{\infty }K(x,t_1,\dots ,t_m){\displaystyle \prod _{j=1}^m}{f}_j(t_j)d{t}_1\dots d{t}_m,$$ where $m\in ?$ and $K:{?}_{+}^{m+1}\to {?}_{+}$ is a continuous kernel function satisfying the condition $$K(x,g_1(x)s_1,\dots ,g_m(x)s_m)=h(x)K(1,s_1,\dots ,s_m),$$ for some functions $g_j,j=\stackrel{￣}{1,m}$, which are continuous, increasing, $g_j({?}_{+})={?}_{+},j=\stackrel{￣}{1,m}$, and a function $h:{?}_{+}\to {?}_{+}$, from a product of weighted-type spaces to weighted-type spaces of real functions. We calculate the norm of the operator, extending and complementing some results in the literature. We also give an explanation for a relation between integrals of an L^p integrable function and its radialization on ${?}^n$.     

Standing waves for a class of fractional p‐Laplacian equations with a general critical nonlinearity

In this paper, we concern with the following fractional p‐Laplacian equation with critical Sobolev exponent $$\left\{\begin{array}{l}{\epsilon }^{ps}{\left\{?\mathrm{\Delta }\right.}_p^{s}u+V(x){\left|u\right|}^{p?2}u=\lambda f(x){\left|u\right|}^{q?2}u+{\left|u\right|}^{p_s^{?}?2}u\text{in}{?}^N,\\ u\in W^{s,p}\left\{{?}^N\right.,u>0,\end{array}\right.$$ where ε > 0 is a small parameter,  λ > 0 , N is a positive integer, and N > ps with s ∈ (0, 1) fixed, $1<q\le p,p_s^{?}:=Np/\left(N?ps\right)$. Since the nonlinearity $h(x,u):=\lambda f(x){\left|u\right|}^{q?2}u+{\left|u\right|}^{p_s^{?}?2}u$ does not satisfy the following Ambrosetti‐Rabinowitz condition: $$0<\mu H(x,u):=\mu {\int }_0^{u}h(x,t)dt\le h(x,u)u,x\in {?}^N,0\ne u\in ?,$$ with μ > p , it is difficult to obtain the boundedness of Palais‐Smale sequence, which is important to prove the existence of positive solutions. In order to overcome the above difficulty, we introduce a penalization method of fractional p‐Laplacian type.     

Addendum to “On the Riesz potential operator of variable order from variable exponent Morrey space to variable exponent Campanato space”, Math Meth Appl Sci. 2020; 1–8

In the paper mentioned in the title, it is proved the boundedness of the Riesz potential operator of variable order α(x) from variable exponent Morrey space to variable exponent Campanato space, under certain assumptions on the variable exponents p(x) and λ(x) of the Morrey space. Assumptions on the exponents were different depending on whether $\frac{\alpha (x)p(x)?n+\lambda (x)}{p(x)}$ takes or not the critical values 0 or 1. In this note, we improve those results by unifying all the cases and covering the whole range $0?\frac{\alpha (x)p(x)?n+\lambda (x)}{p(x)}?1$. We also provide a correction to some minor technicality in the proof of Theorem 2 in the aforementioned paper.     

Two solutions for a class of singular Kirchhoff‐type problems with Hardy–Sobolev critical exponent II

In this article, we devote ourselves to investigate the following singular Kirchhoff‐type equation: $$\left\{\begin{array}{l}?\left(a+b{\int }_{\mathrm{\Omega }}|?u{|}^{2}dx\right)\mathrm{\Delta }u=\frac{u^{5?2s}}{|x{|}^s}+\frac{\lambda }{|x{|}^{\beta }{u}^{\gamma }},x\in \mathrm{\Omega },\\ u>0,x\in \mathrm{\Omega },\\ u=0,x\in ?\mathrm{\Omega },\end{array}\right.$$ where $\mathrm{\Omega }?{?}^3$ is a bounded domain with smooth boundary ?Ω,0∈Ω,a≥0,b,λ>0,0<γ,s<1, and $0\le \beta <\frac{5+\gamma }{2}.$ By using the variational and perturbation methods, we obtain the existence of two positive solutions, which generalizes and improves the recent results in the literature.     

A note on the finite fractal dimension of the global attractors for dissipative nonlinear Schrödinger‐type equations

In this article, we present, throughout two basic models of damped nonlinear Schrödinger (NLS)–type equations, a new idea to bound from above the fractal dimension of the global attractors for NLS‐type equations. This could answer the following open issue: consider, for instance, the classical one‐dimensional cubic nonlinear Schrödinger equation $$u_t+i{u}_{xx}+i|u{|}^{2}u+\gamma u=f,f\in {\mathrm{??}}^2(?).$$ “How can we bound the fractal dimension of the associate global attractor without the need to assume that the external forcing term f has some decay at infinity (that is belonging to some weighted Lebesgue space)?”     

Global weak solutions and eventual smoothness in a 3D two-competing-species chemotaxis-Navier-Stokes system with two consumed signals

This paper deals with a two-competition-species chemotaxis-Navier-Stokes system with two different consumed signals $$\left\{\begin{array}{ll}{(n_1)}_t+\mathbf{u}·\nabla n_1=d_1\mathrm{\Delta }{n}_1-{\chi }_1\nabla ·(n_1\nabla c)+{\mu }_1{n}_1(1-n_1-a_1{n}_2),& \text{in}\mathrm{\Omega }\times (0,\infty ),\\ c_t+\mathbf{u}·\nabla c=d_2\mathrm{\Delta }c-{\alpha }_{1}c{n}_2,& \text{in}\mathrm{\Omega }\times (0,\infty ),\\ {(n_2)}_t+\mathbf{u}·\nabla n_2=d_3\mathrm{\Delta }{n}_2-{\chi }_2\nabla ·(n_2\nabla v)+{\mu }_2{n}_2(1-a_2{n}_1-n_2),& \text{in}\mathrm{\Omega }\times (0,\infty ),\\ v_t+\mathbf{u}·\nabla v=d_4\mathrm{\Delta }v-{\alpha }_{2}v{n}_1,& \text{in}\mathrm{\Omega }\times (0,\infty ),\\ {\mathbf{u}}_t+(\mathbf{u}·\nabla )\mathbf{u}=\mathrm{\Delta }\mathbf{u}+\nabla P+({\beta }_1{n}_1+{\beta }_2{n}_2)\nabla \varphi ,& \text{in}\mathrm{\Omega }\times (0,\infty ),\\ \nabla ·\mathbf{u}=0,& \text{in}\mathrm{\Omega }\times (0,\infty ),\end{array}\right.$$ in a smooth bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^3$ under zero Neumann boundary conditions for $n_1,n_2,c,v$, and homogeneous Dirichlet boundary condition for $\mathbf{\text{u}}$, where the parameters $d_i$ ( $i=1,2,3,4$) and ${\chi }_j,{\mu }_j,a_j,{\alpha }_j,{\beta }_j$ ( $j=1,2$) are positive. This system describes the evolution of two-competing species which react on two different chemical signals in a liquid surrounding environment. Recently, the boundedness and stabilization of classical solutions to the above system under two-dimensional case have been derived in the previous works. However, to the best of our knowledge, the well-posedness problem of solutions for the above system is still open in the three dimensional setting, because of the difficulties in the Navier-Stokes system. The aim of this paper is to construct global weak solutions and show that after some waiting time, these weak solutions become eventually smooth.     

Multibump solutions for nonlinear Schrödinger-Poisson systems

In this paper, we study the following Schrödinger-Poisson equations: $$\left\{\begin{array}{ll}-{\epsilon }^2\mathrm{\Delta }u+V(x)u+K(x)\varphi u=|u{|}^{p-2}u,& x\in {\mathbb{R}}^3,\\ -{\epsilon }^2\mathrm{\Delta }\varphi =K(x)u^2,& x\in {\mathbb{R}}^3,\end{array}\right.$$ where $p\in (4,6)$, $\epsilon >0$ is a parameter and $V$ and $K$ satisfy the critical frequency conditions. By using variational methods and penalization arguments, we show the existence of multibump solutions for the above system. Furthermore, the heights of these bumps are different order.     

Positive ground state solutions for Schrödinger–Poisson system with critical nonlocal term in ℝ3

This paper is dedicated to studying the following Schrödinger–Poisson system $$\left\{\begin{array}{ll}-\mathrm{\Delta }u+V(x)u-K(x)\varphi |u{|}^{3}u=a(x)f(u),& x\in {ℝ}^3,\\ -\mathrm{\Delta }\varphi =K(x)|u{|}^5,& x\in {ℝ}^3.\end{array}\right.$$ Under some different assumptions on functions V(x), K(x), a(x) and f(u), by using the variational approach, we establish the existence of positive ground state solutions.     

A reduced basis approach to large‐scale pseudospectra computation

For studying spectral properties of a nonnormal matrix $A\in {\mathbb{C}}^{n\times n}$, information about its spectrum σ(A) alone is usually not enough. Effects of perturbations on σ(A) can be studied by computing ε‐pseudospectra, i.e. the level sets of the resolvent norm function $\mathsf{g}(z)=\Vert {(zI?A)}^{?1}{\Vert }_2$. The computation of ε‐pseudospectra requires determining the smallest singular values ${\sigma }_{\min }(zI?A)$ for all z on a portion of the complex plane. In this work, we propose a reduced basis approach to pseudospectra computation, which provides highly accurate estimates of pseudospectra in the region of interest, in particular, for pseudospectra estimates in isolated parts of the spectrum containing few eigenvalues of A. It incorporates the sampled singular vectors of zI ? A for different values of z, and implicitly exploits their smoothness properties. It provides rigorous upper and lower bounds for the pseudospectra in the region of interest. In addition, we propose a domain splitting technique for tackling numerically more challenging examples. We present a comparison of our algorithms to several existing approaches on a number of numerical examples, showing that our approach provides significant improvement in terms of computational time.     

On pointwise decay rates of time‐periodic solutions to the Navier–Stokes equation

We study the existence of a time‐periodic solution with pointwise decay properties to the Navier–Stokes equation in the whole space. We show that if the time‐periodic external force is sufficiently small in an appropriate sense, then there exists a time‐periodic solution $\{u,p\}$ of the Navier–Stokes equation such that $|{?}^j{u(t,x)|=O(|x|}^{1?n?j})$ and $|{?}^j{p(t,x)|=O(|x|}^{?n?j})(j=0,1,\dots )$ uniformly in $t\in \mathbb{R}$ as $|x|\to \infty$. Our solution decays faster than the time‐periodic Stokes fundamental solution and the faster decay of its spatial derivatives of higher order is also described.     

Ground state solutions for a modified fractional Schrödinger equation with critical exponent

In this paper, we study the existence of ground state solutions for the modified fractional Schrödinger equations $${(-\mathrm{\Delta })}^{\alpha }u+\mu u+\kappa [{(-\mathrm{\Delta })}^{\alpha }{u}^2]u=\sigma |u{|}^{p-1}u+|u{|}^{q-2}u,x\in R^N,$$ where $N\ge 2$, $\alpha \in (0,1)$, $\mu$, $\sigma$ and $\kappa$ are positive parameters, $2<p+1<q\le 2_{\alpha }^{\ast }:=\frac{2N}{N-2\alpha }$, ${(-\mathrm{\Delta })}^{\alpha }$ denotes the fractional Laplacian of order $\alpha$. For the case $2<p+1<q<2_{\alpha }^{\ast }$ and the case $q=2_{\alpha }^{\ast }$, the existence results of ground state solutions are given, respectively.     

Existence of ground-state solutions for p-Choquard equations with singular potential and doubly critical exponents

We are concerned with the following Choquard equation: $$-{\mathrm{\Delta }}_{p}u+\frac{A}{{|x|}^{\theta }}{|u|}^{p-2}u=\left(I_{\alpha }\ast F(u)\right)f(u),x\in {\mathbb{R}}^N,$$ where $p\in (1,N)$, $\alpha \in (0,N)$, $\theta \in [0,p)\cup \left(p,\frac{(N-1)p}{p-1}\right)$, $A>0$, ${\mathrm{\Delta }}_p$ is the p-Laplacian, $I_{\alpha }$ is the Riesz potential, and F is the primitive of f which is of critical growth due to the Hardy–Littlewood–Sobolev inequality. Under different range of θ and almost necessary conditions on the nonlinearity f in the spirit of Berestycki–Lions-type conditions, we divide this paper into three parts. By applying the refined Sobolev inequality with Morrey norm and the generalized version of the Lions-type theorem, some existence results are established. It is worth noting that our method is not involving the concentration–compactness principle.     

Bounded sets structure of CpX and quasi‐(DF)‐spaces

For wide classes of locally convex spaces, in particular, for the space $C_p(X)$ of continuous real‐valued functions on a Tychonoff space X equipped with the pointwise topology, we characterize the existence of a fundamental bounded resolution (i.e., an increasing family of bounded sets indexed by the irrationals which swallows the bounded sets). These facts together with some results from Grothendieck's theory of $(DF)$‐spaces have led us to introduce quasi‐ $(DF)$‐spaces, a class of locally convex spaces containing $(DF)$‐spaces that preserves subspaces, countable direct sums and countable products. Regular $(LM)$‐spaces as well as their strong duals are quasi‐ $(DF)$‐spaces. Hence the space of distributions $D^{\prime }(\mathrm{\Omega })$ provides a concrete example of a quasi‐ $(DF)$‐space not being a $(DF)$‐space. We show that $C_p(X)$ has a fundamental bounded resolution if and only if $C_p(X)$ is a quasi‐ $(DF)$‐space if and only if the strong dual of $C_p(X)$ is a quasi‐ $(DF)$‐space if and only if X is countable. If X is metrizable, then $C_k(X)$ is a quasi‐ $(DF)$‐space if and only if X is a σ‐compact Polish space.     

Lp‐Estimates for the ∂¯b‐equation on a class of infinite type domains

We prove $L^p$ estimates, $1\le p\le \infty$, for solutions to the tangential Cauchy–Riemann equations ${\overline{?}}_{b}u=?$ on a class of infinite type domains $\mathrm{\Omega }?{\mathbb{C}}^2$. The domains under consideration are a class of convex ellipsoids, and we show that if $?$ is a ${\overline{?}}_b$‐closed (0,1)‐form with coefficients in $L^p$, then there exists an explicit solution u satisfying ${\parallel u\parallel }_{L^p(b\mathrm{\Omega })}\le C{\parallel ?\parallel }_{L^p(b\mathrm{\Omega })}$. Moreover, when $p=\infty$, we show that there is a gain in regularity to an f‐Hölder space. We also present two applications. The first is a solution to the $\overline{?}$‐equation, that is, given a smooth (0,1)‐form ? on $b\mathrm{\Omega }$ with an L¹‐boundary value, we can solve the Cauchy–Riemann equation $\overline{?}u=?$ so that ${\parallel u\parallel }_{L^1(b\mathrm{\Omega })}\le C{\parallel ?\parallel }_{L^1(b\mathrm{\Omega })}$ where C is independent of $u$ and ?. The second application is a discussion of the zero sets of holomorphic functions with zero sets of functions in the Nevanlinna class within our class of domains.