Search

In this paper we study the regularity of weak solutions to an elliptic-parabolic system modeling natural network formation. The system is singular and involves cubic nonlinearity. Our investigation reveals that weak solutions are Hölder continuous when the space dimension $N$ is $2$. This is achieved via an inequality associated with the Stummel-Kato class of functions and refinement of a lemma originally due to S. Campanato and C. B. Morrey (\cite{G}, p. 86).

Pivalic anhydride is shown to be an effective reagent for direct amidation of carboxylic acids with N-alkyl anilines under base-free conditions.

A novel and unprecedented method for the polychlorinated alkylation annulation of N-arylacrylamide has been developed under electrochemical conditions with diazo salt as the mediator at the cathode reduction and CH2Cl2 as the substrate and solvent.

A novel quinone-based imide conjugated microporous polymer-reductive graphene oxide (PIBQ-1%rGO) composite was designed and successfully prepared, and exhibits a high specific capacity of 340 C g−1 at 1 A g−1 in sulfuric acid electrolyte.

A photoredox trifluoromethylation/cyclization of N-arylacrylamides with inexpensive stable solid 4-(difluoroamino)-1-[(trifluoromethyl)sulfonyl]pyridinium triflate (TFSP) has been developed.

We report a catalyst-free method for the alkylation/annulation of N-arylacrylamides to give the corresponding 3-alkyl-2-oxo-2,3-dihydro-1H-indoles by using alkyl iodides as precursors of alkyl radicals. These reactions occur at moderate temperatures and are mediated by easily accessible K2S2O8 and Bu3N.

A visible-light photocatalytic generation of iminoxyl radicals has been accomplished under oxidant-free conditions. This approach offers a mild, atom-economical, and redox-neutral synthesis of 5-methylisoxazolines through radical hydroxygenation of β,γ-unsaturated oximes in the absence of an additional hydrogen source.

The direct hydrogenation reduction of aromatic nitro compounds to aromatic amines with the non-noble metals is an attractive area. Herein, the pyrolysis of Co(2-methylimidazole)2 metal-organic framework successfully produces a magnetic Co-N/C nanocomposite, which exhibits porous structure with high specific area and uniform Co nanoparticles distribution in nitrogen-doped graphite. What’s more, the Co-N/C catalysts possess high cobalt content (23%) with highly active β-Co as the main existing form and high nitrogen content (3%). These interesting characters enable the Co-N/C nanocomposite as the catalyst for the hydrogenation reduction of nitro compounds with excellent catalytic activity under mild condition. In addition, the obtained Co-N/C nanocomposites possess a broad substrate scope and good cycle stability for the reduction of halogen-substituted or carbonyl substituted phenyl nitrates.

The formation of free radicals using visible light as the energy source is of great synthetic value. Despite advances in the photoredox catalysis process using expensive Ir and Ru complexes or organic dyes, the discovery and employment of photocatalysts based on alternative, abundant, and inexpensive metal salts remain highly desired. Herein, the visible-light-induced decarboxylation of α-imino-oxy acids for the generation of iminyl radicals has been accomplished using nontoxic and inexpensive CeCl₃·7H₂O as the photocatalyst via the LMCT process. The iminyl-radical-mediated C–C cleavage/amination and alkene iminoamination have been achieved within this catalyst system.

An iminyl-radical-mediated cyanoalkylarylation of α,β-unsaturated imides with cyclic α-imino oxy acids leading to isoquinoline-1,3(2H,4H)-dione derivatives has been developed. The procedure involves the generation of iminyl radicals through silver-catalyzed oxidative decarboxylation, followed by a C–C bond cleavage, cyanoalkylation, and C–H-functionalization cascade.

We study a class of fourth order nonlinear parabolic equations which include the thin-film equation and the quantum drift-diffusion model as special cases. We investigate these equations by first developing functional inequalities of the type $ \int_\Omega u^{2\gamma-\alpha-\beta}\Delta u^\alpha\Delta u^\beta dx \geq c\int_\Omega|\Delta u^\gamma |^2dx $, which seem to be of interest on their own right.

In this paper we prove the global existence of a strong solution to the initial boundary value problem for the exponential partial differential equation ∂ t u − Δ e − Δ u + e − Δ u − 1 = 0 . The equation was proposed as a continuum model for epitaxial growth of crystal surfaces on vicinal surfaces with evaporation and deposition effects [6] . Our investigations reveal that we must control the size of both ‖ e − Δ u ( x , 0 ) ‖ W 2 , 2 ( Ω ) and ‖ e Δ u ( x , 0 ) ‖ ∞ , Ω suitably to achieve our results. Related results in [8] , [10] were established via the Weiner algebra framework. Here we offer a totally new approach, which seems to shed more light on the nature of exponential nonlinearity.

A mild and metal-free procedure is reported for the aerobic oxidation of substituted toluenes to carboxylic acids by using CBr4 as initiator under irradiation from a 400 nm blue light-emitting diode.

It is well known that a weak solution φ to the initial boundary value problem for the uniformly parabolic equation $\partial _t\varphi - {\rm div}(A\nabla \varphi ) +\omega \varphi = f $ in $\Omega _T\equiv \Omega \times (0,T)$ satisfies the uniform estimate $$\Vert \varphi \Vert_{\infty,\Omega_T}\les \Vert \varphi\Vert_{\infty,\partial_p\Omega_T}+c \Vert f \Vert_{q,\Omega_T}, \ \ \ c=c(N,\lambda, q, \Omega_T), $$ provided that $q \gt 1+{N}/{2}$, where Ω is a bounded domain in ${\open R}^N$ with Lipschitz boundary, T > 0, $\partial _p\Omega _T$ is the parabolic boundary of $\Omega _T$, $\omega \in L^1(\Omega _T)$ with $\omega \ges 0$, and λ is the smallest eigenvalue of the coefficient matrix A. This estimate is sharp in the sense that it generally fails if $q=1+{N}/{2}$. In this paper, we show that the linear growth of the upper bound in $\Vert f \Vert_{q,\Omega _T}$ can be improved. To be precise, we establish $$ \Vert \varphi \Vert_{\infty,\Omega_T}\les \Vert \varphi_0 \Vert_{\infty,\partial_p\Omega_T}+c \Vert f \Vert_{1+{N}/{2},\Omega_T} \left(\ln(\Vert f \Vert_{q,\Omega_T}+1)+1\right). $$

An efficient intramolecular iodooxygenation of unsaturated ketoximes has been achieved through the activation of iodine by TBHP, yielding various types of synthetically versatile iodinated isoxazol...

We study regularity properties of weak solutions to the boundary value problem for the equation −Δρ + au = f in a bounded domain Ω ⊂ R N , where ρ = e − div | ∇ u | p − 2 ∇ u + β 0 | ∇ u | − 1 ∇ u . This problem is derived from the mathematical modelling of crystal surfaces. It is known that the exponential term can be a measure-valued function. In this paper we obtain a partial regularity result, which asserts that there exists an open subset Ω0 ⊂ Ω such that |Ω\Ω0| = 0 and the exponential term is locally bounded in Ω0. Furthermore, if x 0 ∈ Ω\Ω0, then ρ vanishes of N + 2 − ɛ order at x 0 for each ɛ ∈ (0, 2).

We investigate the possible blow-up of strong solutions to a biological network formation model originally introduced by D. Cai and D. Hu \cite{HC}. The model is represented by an initial boundary value problem for an elliptic-parabolic system with cubic non linearity. We obtain an algebraic equation for the possible blow-up time of strong solutions. The equation yields information on how various given data may contribute to the blow-up of solutions. As a by-product of our development, we establish a $W^{1,q}$ estimate for solutions to an elliptic equation which shows the explicit dependence of the upper bound on the elliptic coefficients.

In this paper we study the boundary value problem for the equation $$\text{ div }\left( D(\nabla u)\nabla \left( \text{ div }\left( |\nabla u|^{p-2}\nabla u+\beta \frac{\nabla u}{|\nabla u|}\right) \right) \right) +au=f$$ in the $$z=(x,y)$$ plane. This problem is derived from a continuum model for the relaxation of a crystal surface below the roughing temperature. The mathematical challenge is of twofolds. First, the mobility $$D(\nabla u)$$ is a $$2\times 2$$ matrix whose smallest eigenvalue is not bounded away from 0 below. Second, the equation contains the 1-Laplace operator, whose mathematical properties are still not well-understood. Existence of a weak solution is obtained. In particular, $$|\nabla u|$$ is shown to be bounded when $$p>\frac{4}{3}$$ .

A complementary and general strategy for the oxidative generation of iminyl radicals from the readily available α-imino-oxy acids has been established through silver-catalyzed decarboxylation. To demonstrate its synthesis utility, the direct C-H cyanoalkylation of heterocycles and quinones with cyclic α-imino-oxy acids via the iminyl radical-mediated C-C bond cleavage is developed. This cost-effective method takes place under mild reaction conditions and exhibits a broad substrate scope.

In this paper we study the initial boundary value problem for the system u t − Δ u m = − div ( u q ∇ v ) , v t − Δ v + v = u . This problem is the so-called Keller-Segel model with nonlinear diffusion. Our investigation reveals that nonlinear diffusion can prevent overcrowding. To be precise, we show that solutions are bounded as long as m > q > 0 , thereby substantially generalizing the known results in this area. Furthermore, our result seems to imply that the Keller-Segel model can have bounded solutions and blow-up ones simultaneously.

This article is concerned with the existence of a weak solution to the initial boundary problem for a cross-diffusion system which arises in the study of two cell population growth. The mathematical challenge is due to the fact that the coefficient matrix is non-symmetric and degenerate in the sense that its determinant is \begin{document}$ 0 $\end{document} . The existence assertion is established by exploring the fact that the total population density satisfies a porous media equation.

In this paper, we study the initial boundary value problem for the system $$\Delta v= u_{x_1},\ u_t-\text{ div }\left( \left( (a|{\mathbf {q}}|+m)I+(b-a)\frac{{\mathbf {q}}\otimes {\mathbf {q}}}{|{\mathbf {q}}|}\right) \nabla u\right) =-\nabla u\cdot {\mathbf {q}}$$ , where $${\mathbf {q}}=(-v_{x_2}, v_{x_1})^{\mathrm{T}}, {\mathbf {q}}\otimes {\mathbf {q}}={\mathbf {q}}{\mathbf {q}}^{\mathrm{T}}$$ . This problem has been proposed as a model for a fluid flowing through a porous medium under the influence of gravity and hydrodynamic dispersion. For each $$T>0$$ , we obtain a so-called strong solution (v, u) in the function space $$L^\infty (0,T; \left( W^{1,\infty }(\Omega )\right) ^2)$$ , where $$\Omega $$ is a bounded domain in $${\mathbb {R}}^2$$ . The key ingredient in our approach is the decomposition $$A^2=\text{ tr } (A)A-\text{ det }(A) I$$ for any $$2\times 2$$ symmetric matrix A. By exploring this decomposition, we are able to derive an equation of parabolic type for the function $$\left( \left( (a|{\mathbf {q}}|+m)I+(b-a)\frac{{\mathbf {q}}\otimes {\mathbf {q}}}{|{\mathbf {q}}|}\right) \nabla u\cdot \nabla u\right) ^j, j\ge 1$$ . With the aid of this equation, we obtain a uniform bound for $$\nabla u$$ .

In this paper we study the initial boundary value problem for the system \begin{document}$ -\mbox{div}\left[(I+\mathbf{m} \mathbf{m}^T)\nabla p\right] = s(x), \ \ \mathbf{m}_t-\alpha^2\Delta\mathbf{m}+|\mathbf{m}|^{2(\gamma-1)}\mathbf{m} = \beta^2(\mathbf{m}\cdot\nabla p)\nabla p $\end{document} in two space dimensions. This problem has been proposed as a continuum model for biological transportation networks. The mathematical challenge is due to the presence of cubic nonlinearities, also known as trilinear forms, in the system. We obtain a weak solution \begin{document}$ (\mathbf{m}, p) $\end{document} with both \begin{document}$ |\nabla p| $\end{document} and \begin{document}$ |\nabla\mathbf{m}| $\end{document} being bounded. The result immediately triggers a bootstrap argument which can yield higher regularity for the weak solution. This is achieved by deriving an equation for \begin{document}$ v\equiv(I+\mathbf{m} \mathbf{m}^T)\nabla p\cdot\nabla p $\end{document} , and then suitably applying the De Giorge iteration method to the equation.

An unexpected regioselective I2/TBHP-mediated 1,2-iodo­oxygenation of alkenes with N-hydroxylamines is described. The reaction proceeds through a radical coupling mechanism or a iodonium mechanism, which is controlled by the structures of both N-hydroxylamines and alkenes, to form vicinal iodo-substituted N-alkoxyamines regioselectively. Both the iodo and alkoxyamine group of the resulting products offer rich possibilities of synthetic manipulations.

In this paper we investigate regularity properties of weak solutions to a PDE system that arises in the study of biological transport networks. The system consists of a possibly singular elliptic equation for the scalar pressure of the underlying biological network coupled to a diffusion equation for the conductance vector of the network. There are several different types of nonlinearities in the system. Of particular mathematical interest is a term that is a polynomial function of solutions and their partial derivatives and this polynomial function has degree three. That is, the system contains a cubic nonlinearity. Only weak solutions to the system have been shown to exist. The regularity theory for the system remains fundamentally incomplete. In particular, it is not known whether or not weak solutions develop singularities. In this paper we obtain a partial regularity theorem, which gives an estimate for the parabolic Hausdorff dimension of the set of possible singular points.

A Sc(OTf)3-catalyzed aza-Friedel–Crafts reaction of electron-rich arenes with isatin-derived ketimines is disclosed, providing biologically interesting 3-amino-3-aryl-2-oxindoles in moderate to good yields with broad substrate scope. This reaction can be easily scaled up, and the Boc protecting group could be readily removed under mild reaction conditions to afford the free amine.