Recent zbMATH articles in MSC 35C15https://zbmath.org/atom/cc/35C152021-06-15T18:09:00+00:00WerkzeugBlow-up for the 1D nonlinear Schrödinger equation with point nonlinearity. II: Supercritical blow-up profiles.https://zbmath.org/1460.353282021-06-15T18:09:00+00:00"Holmer, Justin"https://zbmath.org/authors/?q=ai:holmer.justin"Liu, Chang"https://zbmath.org/authors/?q=ai:liu.chang.1Summary: We consider the 1D nonlinear Schrödinger equation (NLS) with focusing point nonlinearity,
\[
i\partial_t\psi+\partial_x^2\psi+\delta|\psi|^{p-1}\psi=0,\tag{(0.1)}
\]
where \(\delta=\delta(x)\) is the delta function supported at the origin. In the \(L^2\) supercritical setting \(p>3\), we construct self-similar blow-up solutions belonging to the energy space \(L_x^\infty\cap\dot H_x^1\). This is reduced to finding outgoing solutions of a certain stationary profile equation. All outgoing solutions to the profile equation are obtained by using parabolic cylinder functions (Weber functions) and solving the jump condition at \(x=0\) imposed by the \(\delta\) term in (0.1). This jump condition is an algebraic condition involving gamma functions, and existence and uniqueness of solutions is obtained using the intermediate value theorem and formulae for the digamma function. We also compute the form of these outgoing solutions in the slightly supercritical case \(0<p-3\ll 1\) using the log Binet formula for the gamma function and steepest descent method in the integral formulae for the parabolic cylinder functions.
For part I, see [the authors, J. Math. Anal. Appl. 483, No. 1, Article ID 123522, 20 p. (2020; Zbl 1436.35290)].Periodic solutions of an age-structured epidemic model with periodic infection rate.https://zbmath.org/1460.353532021-06-15T18:09:00+00:00"Kang, Hao"https://zbmath.org/authors/?q=ai:kang.hao"Huang, Qimin"https://zbmath.org/authors/?q=ai:huang.qimin"Ruan, Shigui"https://zbmath.org/authors/?q=ai:ruan.shiguiSummary: In this paper we consider an age-structured epidemic model of the susceptible-exposed-infectious-recovered (SEIR) type. To characterize the seasonality of some infectious diseases such as measles, it is assumed that the infection rate is time periodic. After establishing the well-posedness of the initial-boundary value problem, we study existence of time periodic solutions of the model by using a fixed point theorem. Some numerical simulations are presented to illustrate the obtained results.Parabolic equations involving Laguerre operators and weighted mixed-norm estimates.https://zbmath.org/1460.350662021-06-15T18:09:00+00:00"Fan, Huiying"https://zbmath.org/authors/?q=ai:fan.huiying"Ma, Tao"https://zbmath.org/authors/?q=ai:ma.taoSummary: In this paper, we study evolution equation \(\partial_tu=-L_\alpha u+f\) and the corresponding Cauchy problem, where \(L_\alpha\) represents the Laguerre operator \(L_\alpha=\frac{1}{2}(-\frac{d^2}{dx^2}+x^2+\frac{1}{x^2}(\alpha^2-\frac{1}{4}))\), for every \(\alpha\geq-\frac{1}{2}\). We get explicit pointwise formulas for the classical solution and its derivatives by virtue of the parabolic heat-diffusion semigroup \(\{e^{-\tau(\partial_t+L_\alpha)}\}_{\tau>0}\). In addition, we define the Poisson operator related to the fractional power \((\partial_t+L_\alpha)^s\) and reveal weighted mixed-norm estimates for revelent maximal operators.Representation of solutions of the Cauchy problem for a one-dimensional Schrödinger equation with a smooth bounded potential by quasi-Feynman formulae.https://zbmath.org/1460.810172021-06-15T18:09:00+00:00"Grishin, Denis V."https://zbmath.org/authors/?q=ai:grishin.denis-v"Pavlovskiy, Yan Yu."https://zbmath.org/authors/?q=ai:pavlovskiy.yan-yuHardy inequalities for the fractional powers of the Grushin operator.https://zbmath.org/1460.350072021-06-15T18:09:00+00:00"Song, Manli"https://zbmath.org/authors/?q=ai:song.manli"Tan, Jinggang"https://zbmath.org/authors/?q=ai:tan.jinggangSummary: We establish uncertainty principles and Hardy inequalities for the fractional Grushin operator, which are reduced to those inequalities for the fractional generalized sublaplacian. The key ingredients to obtain them are an explicit integral representation and a ground state representation for the fractional powers of generalized sublaplacian.On properties of solutions to the \(\alpha\)-harmonic equation.https://zbmath.org/1460.310022021-06-15T18:09:00+00:00"Li, Peijin"https://zbmath.org/authors/?q=ai:li.peijin"Rasila, Antti"https://zbmath.org/authors/?q=ai:rasila.antti"Wang, Zhi-Gang"https://zbmath.org/authors/?q=ai:wang.zhigangSummary: The aim of this paper is to establish properties of solutions to the \(\alpha \)-harmonic equations: \(\Delta_\alpha(f(z))=\partial z[(1-|z|^2)^{-\alpha}\bar{\partial}zf](z)=g(z)\), where \(g:\overline{\mathbb{D}}\to\mathbb{C}\) is a continuous function and \(\overline{\mathbb{D}}\) denotes the closure of the unit disc \(\mathbb{D}\) in the complex plane \(\mathbb{C}\). We obtain Schwarz type and Schwarz-Pick type inequalities for solutions to the \(\alpha\)-harmonic equation. In particular, for \(g\equiv 0\), solutions to the above equation are called \(\alpha\)-harmonic functions. We determine the necessary and sufficient conditions for an analytic function \(\psi\) to have the property that \(f\circ\psi\) is \(\alpha\)-harmonic function for any \(\alpha\)-harmonic function \(f\). Furthermore, we discuss the Bergman-type spaces on \(\alpha\)-harmonic functions.