In this work we study the problem $u_t{-div\left(\right|x|}^{-2\gamma }\nabla u)=\lambda \frac{u^{\alpha }}{{\left|x\right|}^{2(\gamma +1)}}+f$ in $\Omega \times (0,T)$, $u\ge 0$ in $\Omega \times (0,T)$, $u=0$ on $\partial \Omega \times (0,T)$, $u(x,0)=u_0\left(x\right)$ in $\Omega$, $\Omega \subset {ℝ}^N$ $(N\ge 2)$ is a bounded regular domain such that $0\in \Omega$, $\lambda >0$, $\alpha >0$, $-\infty <\gamma <\left(N-2\right)/2$, $f$ and $u_0$ are positive functions such that $f\in L^1(\Omega \times (0,T))$ and $u_0\in L^1\left(\Omega \right)$. The main points under analysis are: (i) spectral instantaneous and complete blow-up related to the Harnack inequality in the case $\alpha =1$, $1+\gamma >0$; (ii) the nonexistence of solutions if $\alpha >1$, $1+\gamma >0$; (iii) a uniqueness result for weak solutions (in the distribution sense); (iv) further results on existence of weak solutions...

We characterise the boundedness of the $H^{\infty }$ calculus of a sectorial operator in terms of dilation theorems. We show e. g. that if $-A$ generates a bounded analytic $C_0$ semigroup $\left(T_t\right)$ on a UMD space, then the $H^{\infty }$ calculus of $A$ is bounded if and only if $\left(T_t\right)$ has a dilation to a bounded group on $L^2([0,1],X)$. This generalises a Hilbert space result of C.LeMerdy. If $X$ is an $L^p$ space we can choose another $L^p$ space in place of $L^2([0,1],X)$.

This paper is concerned with the study of a nonlocal nonlinear parabolic problem associated with the equation $u_t-M\left({\int }_{\Omega }\phi u\mathrm{d}x\right)\mathrm{div}\left(A(x,t,u)\nabla u\right)=g(x,t,u)$ in $\Omega \times (0,T)$, where $\Omega$ is a bounded domain of ${ℝ}^n$ $(n\ge 1)$, $T>0$ is a positive number, $A(x,t,u)$ is an $n\times n$ matrix of variable coefficients depending on $u$ and $M:ℝ\to ℝ$, $\phi :\Omega \to ℝ$, $g:\Omega \times (0,T)\times ℝ\to ℝ$ are given functions. We consider two different assumptions on $g$. The existence of a weak solution for this problem is proved using the Schauder fixed point theorem for each of these assumptions. Moreover, if $A(x,t,u)=a(x,t)$ depends only on...

Given a topological space $X$, let $𝒰X$ and ${\eta }_X:X\to 𝒰X$ denote, respectively, the Salbany compactification of $X$ and the compactification map called the Salbany map of $X$. For every continuous function $f:X\to Y$, there is a continuous function $𝒰f:𝒰X\to 𝒰Y$, called the Salbany lift of $f$, satisfying $\left(𝒰f\right)\circ {\eta }_X={\eta }_Y\circ f$. If a continuous function $f:X\to Y$ has a stably compact codomain $Y$, then there is a Salbany extension $F:𝒰X\to Y$ of $f$, not necessarily unique, such that $F\circ {\eta }_X=f$. In this paper, we give a condition on a space such that its Salbany map is open. In...

We consider problem (P) of Kirchhoff-Carrier type with nonlinear terms containing a finite number of unknown values $u({\eta }_1,t),\cdots ,u({\eta }_q,t)$ with $0\le {\eta }_1<{\eta }_2<\cdots <{\eta }_q<1.$ By applying the linearization method together with the Faedo-Galerkin method and the weak compact method, we first prove the existence and uniqueness of a local weak solution of problem (P). Next, we consider a specific case $\left({\mathrm{P}}_q\right)$ of (P) in which the nonlinear term contains the sum $S_q\left[u^2\right]\left(t\right)=q^{-1}{\sum }_{i=1}^q{u}^2(\frac{(i-1)}{q},t)$. Under suitable conditions, we prove that the solution of $\left({\mathrm{P}}_q\right)$ converges to the solution...

We consider the equation $\mathrm{d}y\left(t\right)/\mathrm{d}t=(A+B(t\left)\right)y\left(t\right)$ $(t\ge 0)$, where $A$ is the generator of an analytic semigroup ${\left({\mathrm{e}}^{At}\right)}_{t\ge 0}$ on a Banach space $𝒳$, $B\left(t\right)$ is a variable bounded operator in $𝒳$. It is assumed that the commutator $K\left(t\right)=AB\left(t\right)-B\left(t\right)A$ has the following property: there is a linear operator $S$ having a bounded left-inverse operator $S_l^{-1}$ such that $\parallel S{\mathrm{e}}^{At}\parallel$ is integrable and the operator $K\left(t\right)S_l^{-1}$ is bounded. Under these conditions an exponential stability test is derived. As an example we consider a coupled system of parabolic equations. ...

Let $D$ be a ${𝒞}^d$ $q$-convex intersection, $d\ge 2$, $0\le q\le n-1$, in a complex manifold $X$ of complex dimension $n$, $n\ge 2$, and let $E$ be a holomorphic vector bundle of rank $N$ over $X$. In this paper, ${𝒞}^k$-estimates, $k=2,3,\cdots ,\infty$, for solutions to the $\overline{\partial }$-equation with small loss of smoothness are obtained for $E$-valued $(0,s)$-forms on $D$ when $n-q\le s\le n$. In addition, we solve the $\overline{\partial }$-equation with a support condition in ${𝒞}^k$-spaces. More precisely, we prove that for a $\overline{\partial }$-closed form $f$ in ${𝒞}_{0,q}^k(X\setminus D,E)$, $1\le q\le n-2$, $n\ge 3$, with compact support and for $\epsilon$ with $0<\epsilon <1$ there...

Let $\mathscr{H}$ be a complex Hilbert space, $A$ a positive operator with closed range in $ℬ\left(\mathscr{H}\right)$ and ${ℬ}_A\left(\mathscr{H}\right)$ the sub-algebra of $ℬ\left(\mathscr{H}\right)$ of all $A$-self-adjoint operators. Assume $\phi :{ℬ}_A\left(\mathscr{H}\right)$ onto itself is a linear continuous map. This paper shows that if $\phi$ preserves $A$-unitary operators such that $\phi \left(I\right)=P$ then $\psi$ defined by $\psi \left(T\right)=P\phi \left(PT\right)$ is a homomorphism or an anti-homomorphism and $\psi \left(T^{♯}\right)=\psi {\left(T\right)}^{♯}$ for all $T\in {ℬ}_A\left(\mathscr{H}\right)$, where $P=A^{+}A$ and $A^{+}$ is the Moore-Penrose inverse of $A$. A similar result is also true if $\phi$ preserves $A$-quasi-unitary operators in both directions such that there...

We investigate the recovery of $k$-sparse signals using the ${\ell }_1$-${\ell }_2$ minimization model with prior support set information. The prior support set information, which is believed to contain the indices of nonzero signal elements, significantly enhances the performance of compressive recovery by improving accuracy, efficiency, reducing complexity, expanding applicability, and enhancing robustness. We assume $k$-sparse signals $𝐱$ with the prior support $T$ which is composed of $g$ true indices and $b$ wrong...