Fractional heat equations with subcritical absorption having a measure as initial data

We study existence and uniqueness of weak solutions to (F) \partial\_t u+ (-\Delta)^\alphau+h(t, u)=0 in $(0,\infty)\times\R^N$,with initial condition $u(0,\cdot)=\nu$ in $\R^N$, where $N\ge2$, the operator $(-\Delta)^\alpha$is the fractional Laplacian with $\alpha\in(0,1)$, $\nu$ isa bounded Radon measure and $h:(0,\infty)\times\R\to\R$ is a continuous function satisfying a subcritical integrability condition.In particular, if $h(t,u)=t^\beta u^p$ with \beta\textgreater{}-1 and 0 \textless{} p \textless{} p^*\_\beta:=1+\frac{2\alpha(1+\beta)}{N}, we prove that there exists a unique weak solution $u\_k$ to (F) with $\nu=k\delta\_0$, where $\delta\_0$ is the Dirac mass at the origin. We obtain that $u\_k\to\infty$ in $(0,\infty)\times\R^N$ as $k\to\infty$ for $p\in(0,1]$ and the limit of $u\_k$ exists as $k\to\infty$ when 1 \textless{} p \textless{} p^*\_\beta, we denote it by $u\_\infty$.When 1+\frac{2\alpha(1+\beta)}{N+2\alpha}:=p^{**}\_\beta\textless{} p \textless{} p^*\_\beta,$u\_\infty$ is the minimal self-similar solution of $(F)\_\infty$ $\partial\_t u+ (-\Delta)^\alpha u+t^\beta u^p=0$ in $(0,\infty)\times\R^N$ with the initial condition $u(0,\cdot)=0$ in $\R^N\setminus\{0\}$ and it satisfies $u\_\infty(0,x)=0$ for $x\neq 0$.While if 1\textless{} p \textless{} p^{**}\_\beta, then $u\_\infty\equiv U\_p$, where $U\_p$ is the maximal solution of the differential equation $y'+t^\beta y^p=0$ on $\R\_+$.Comment: Nonlinear Analysis, Theory, Methods and Applications, to appea

Constraints on anomalous quartic gauge couplings via WγjjW\gamma jj production at the LHC

The vector boson scattering at the Large Hadron Collider (LHC) is sensitive to anomalous quartic gauge couplings (aQGCs). In this paper, we investigate the aQGC contribution to $W \gamma jj$ production at the LHC with $\sqrt{s}=13$ TeV in the context of an effective field theory (EFT). The unitarity bound is applied as a cut on the energy scale of this production process, which is found to have significant suppressive effects on the signals. To enhance the statistical significance, we analyse the kinematic and polarization features of the aQGC signals in detail. We find that the polarization effects induced by the aQGCs are unique and can discriminate the signals from the SM backgrounds well. With the proposed event selection strategy, we obtain the constraints on the coefficients of dimension-8 operators with current luminosity. The results indicate that the process $pp \to W \gamma jj$ is powerful for searching for the $O_{M_{2,3,4,5}}$ and $O_{T_{5,6,7}}$ operators.Comment: 29 pages, 11 figures, 7 tables, to be published in Chinese Physics