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Because of the prevalent time-delay characteristics in real-world phenomena, this paper investigates the existence of mild solutions for diffusion equations with time delays and the Hilfer fractional derivative. This derivative extends the traditional Caputo and Riemann–Liouville fractional derivatives, offering broader practical applications. Initially, we constructed Banach spaces required to handle the time-delay terms. To address the challenge of the unbounded nature of the solution operator at the initial moment, we developed an equivalent continuous operator. Subsequently, within the contexts of both compact and non-compact analytic semigroups, we explored the existence and uniqueness of mild solutions, considering various growth conditions of nonlinear terms. Finally, we presented an example to illustrate our main conclusions.

We primarily investigate the existence of solutions for fractional neutral integro-differential equations with nonlocal initial conditions, which are crucial for understanding natural phenomena. Taking into account factors such as neutral type, fractional-order integrals, and fractional-order derivatives, we employ probability density functions, Laplace transforms, and resolvent operators to formulate a well-defined concept of a mild solution for the specified equation. Following this, by using fixed-point theorems, we establish the existence of mild solutions under more relaxed conditions.

Fractional integrodifferential diffusion equations play a significant role in describing anomalous diffusion phenomena. In this paper, we study the existence and uniqueness of mild solutions to these equations. Firstly, we construct an appropriate resolvent family, through which the related equicontinuity, strong continuity, and compactness properties are studied using the convolution theorem of Laplace transform, the probability density function, the Cauchy integral formula, and the Fubini theorem. Then, we construct a reasonable mild solution for the considered equations. Finally, we obtain some sufficient conditions for the existence and uniqueness of mild solutions to the considered equations by some fixed point theorems.

We obtain a new maximum principle of the periodic solutions when the corresponding impulsive equation is linear. If the nonlinear is quasi-monotonicity, we study the existence of the minimal and maximal periodic mild solutions for impulsive partial differential equations by using the perturbation method, the monotone iterative technique and the method of upper and lower solution. We give an example in last part to illustrate the main theorem.

AIM: To analyze the characteristics of choroidal neovascularization(CNV)in neovascular age-related macular degeneration(nARMD)patients and the outcome of intravitreal anti-vascular endothelial growth factor(VEGF)treatment by using optical coherence tomography angiography(OCTA). METHODS: A prospective cohort study was carried out, which included 37 eyes of 29 patients with nARMD in West China Hospital during May to December 2017. OCTA scans was conducted to all patients before treatment, 1d, 1wk, 1mo and 3-6mo after treatment. The analysis was performed to evaluate the morphological characteristics, lesion area, parafoveal superficial vessel density and perfusion area of CNV before and after treatment. RESULTS: Among all the subjects, immature structure, small branches, and capillaries responded well to anti-VEGF treatment. Compared with the mean lesion area in nARMD patients before treatment(1.27±1.88mm2), there was significant reduction(1.13±1.79 mm2)1d after treatment, which meant CNV lesion decreased 1d after treatment, and stabilized 1mo later(P=0.001). Obvious decrease was observed both in parafoveal superficial vessel density(P=0.003)and perfusion area(P=0.015)3mo after treatment in nARMD patients. CONCLUSION: OCTA, a non-invasive diagnostic examination, clearly identified tiny structures of CNV, quantified the lesion area and displayed specific vasculature in nARMD patients. Furthermore, retinal microcirculation can be detected using OCTA, which provides an effective approach of monitoring the progression and treatment effect of nARMD.

In this paper, a generalized Gronwall inequality is demonstrated, playing an important role in the study of fractional differential equations. In addition, with the fixed-point theorem and the properties of Mittag–Leffler functions, some results of the existence as well as asymptotic stability of square-mean S-asymptotically periodic solutions to a fractional stochastic diffusion equation with fractional Brownian motion are obtained. In the end, an example of numerical simulation is given to illustrate the effectiveness of our theory results.

In this paper, we investigate a class of nonautonomous fractional diffusion equations (NFDEs). Firstly, under the condition of weighted Hölder continuity, the existence and two estimates of classical solutions are obtained by virtue of the properties of the probability density function and the evolution operator family. Secondly, it focuses on the continuity and an estimate of classical solutions in the sense of fractional power norm. The results generalize some existing results on classical solutions and provide theoretical support for the application of NFDE.

In this article, we consider the stability for elliptic problems that have degenerate coercivity in their principal part, $$\displaylines{ -\text{div}\Big(\frac{|\nabla u|^{p-2}\nabla u}{(1+|u|)^{\theta(p-1)}}\Big) +|u|^{q-1}u=f,\quad x\in\Omega, \cr u(x)=0,\quad x\in \partial\Omega, }$$ where $\theta>0$, $\Omega\subseteq \mathbb{R}^N$ is a bounded domain. Let K be a compact subset in $\Omega$ with zero r-capacity ($pr(p-1)[1+\theta(p-1)]/(r-p)$, and $u_n$ is the sequence of solutions of the corresponding problems with datum $f_n$. Then $u_n$ converges to the solution u.

This paper deals with the existence and uniqueness of periodic solutions, S-asymptotically periodic solutions, and other types of bounded solutions for some fractional evolution equations with the Weyl-Liouville fractional derivative defined for periodic functions. Applying Fourier transform we give reasonable definitions of mild solutions. Then we accurately estimate the spectral radius of resolvent operator and obtain some existence and uniqueness results.

Local integral estimates as well as local nonexistence results for a class of quasilinear equations -Δpu=σP(u)+ω for p>1 and Hessian equations Fk-u=σP(u)+ω were established, where σ is a nonnegative locally integrable function or, more generally, a locally finite measure, ω is a positive Radon measure, and P(u)~exp⁡αuβ with α>0 and β≥1 or P(u)=up-1.

We use a monotone iterative method in the presence of lower and upper solutions to discuss the existence and uniqueness of mild solutions for the initial value problem $$displaylines{ u'(t)+Au(t)= f(t,u(t),Tu(t)),quad tin J,; t eq t_k,cr Delta u |_{t=t_k}=I_k(u(t_k)) ,quad k=1,2,dots ,m,cr u(0)=x_0, }$$ where $A:D(A)subset Eo E$ is a closed linear operator and $-A$ generates a strongly continuous semigroup $T(t)(tgeq 0)$ in $E$. Under wide monotonicity conditions and the non-compactness measure condition of the nonlinearity f, we obtain the existence of extremal mild solutions and a unique mild solution between lower and upper solutions requiring only that $-A$ generate a strongly continuous semigroup.

The periodic boundary value problem is discussed for a class of fractional evolution equations. The existence and uniqueness results of mild solutions for the associated linear fractional evolution equations are established, and the spectral radius of resolvent operator is accurately estimated. With the aid of the estimation, the existence and uniqueness results of positive mild solutions are obtained by using the monotone iterative technique. As an application that illustrates the abstract results, an example is given.

We mainly study the fractional evolution equation in an ordered Banach space 𝑋𝐶𝐷𝑎0+𝑢(𝑡)+𝐴𝑢(𝑡)=𝑓(𝑡,𝑢(𝑡),𝐺𝑢(𝑡)) , 1

We investigate the initial value problem for a class of fractional evolution equations in a Banach space. Under some monotone conditions and noncompactness measure conditions of the nonlinearity, the well-known monotone iterative technique is then extended for fractional evolution equations which provides computable monotone sequences that converge to the extremal solutions in a sector generated by upper and lower solutions. An example to illustrate the applications of the main results is given.

Oscillation characteristics in a cavity are investigated under real experimental conditions through unsteady numerical simulations of the time-evolving oscillatory damping of a high-Mach-number freestream over a two-dimensional forward-facing cavity. The post-disturbance flow field is taken as the initial condition. Temporal variations in the flow field and wall resistance coefficient are obtained. The forward-facing cavity experiences underdamped oscillatory behavior when subjected to disturbances. The convergence of the oscillations is influenced by the cavity volume, with significant reductions in cavity damping observed when stagnation regions develop within the cavity. During the initial phase of disturbance, each oscillation cycle consists of gas injection and jet phases. In the former, external gas stagnates within the cavity, resulting in a gradual increase in internal density and pressure. High-temperature regions extend from the external flow into the cavity, and bow shocks approach the cavity wall, adversely affecting aerodynamic drag reduction and thermal protection for aircrafts. In the jet phase, the flow field structure resembles the opposing jet. As the gas is expelled, the internal cavity pressure decreases, forming a cold jet that envelops the cavity's surface. The temperature within the boundary layer on the surface decreases, and bow shocks are pushed away from the wall, resulting in thermal-protection and drag-reduction effects. Transitions between phases induce instability in the internal flow states within the cavity. During the transition from the gas injection phase to the jet phase, the wall drag coefficient reaches its peak value; the reverse transition results in the lowest wall drag coefficient. [ABSTRACT FROM AUTHOR]

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Wireless localization and target sensing both rely on precise extraction of parameters such as signal amplitude, propagation delay, and Doppler shift from the received signals. Due to the high multi-path resolution and strong penetration, both localization and sensing can be achieved through identical UWB waveforms. In this paper, we try to properly allocate resources for localization and sensing to fully exploit the potential of UWB systems. Considering the complexity of the multi-slot networks, we derive the Fisher Information Matrix (FIM) expressions for single-slot and dual-slot integrated sensing and localization (ISAL) networks, respectively, and propose two resource optimization schemes, namely the step-by-stepscheme and the integrated scheme, respectively. The numerical results show that: (i) for the sensing-resource-limited networks with relatively uniform node distribution, the energy allocated to each step in the step-by-step scheme satisfies the relationship energy for clock offset < energy for radar localization < energy for target sensing; (ii) in the multi-slot ISAL networks, more energy will be allocated to the time slots where the networks are relatively sensing-resource-limited; (iii) the step-by-step scheme is more suitable for the sensing-resource-abundant networks, while the integrated scheme is more suitable for the sensing-resource-limited networks. [ABSTRACT FROM AUTHOR]