Acta Applicandae Mathematicae最新文献

We consider the stability of the global weak solution of the Quantum Euler system in two space dimensions. More precisely, we establish compactness properties of global finite energy weak solution for large initial data provided that these are axisymmetric. The main novelty is that the initial velocity is not necessary irrotational when the density is not vanishing, our main argument is based on the Madelung transform which enables us to prove new Kato estimates on the irrotational part of the velocity.

The non-linear collision-induced breakage equation has significant applications in particulate processes. Two semi-analytical techniques, namely homotopy analysis method (HAM) and accelerated homotopy perturbation method (AHPM) are investigated along with the well-known numerical scheme finite volume method (FVM) to comprehend the dynamical behavior of the non-linear system, i.e., the concentration function, the total number, total mass and energy dissipation of the particles in the system. These semi-analytical methods provide approximate analytical solutions by truncating the infinite series form. The theoretical convergence analyses of the series solutions of HAM and AHPM are discussed under some assumptions on the collisional kernels. In addition, the error estimations of the truncated solutions of both methods equip the maximum absolute error bound. Moreover, HAM simulations are computationally costly compared to AHPM because of an additional auxiliary parameter. To justify the applicability and accuracy of these series methods, approximated solutions are compared with the findings of FVM and analytical solutions considering three physical problems.

In this paper, a critical fourth-order Kirchhoff type elliptic equation with a subcritical perturbation is studied. The main feature of this problem is that it involves both a nonlocal coefficient and a critical term, which bring essential difficulty for the proof of the existence of weak solutions. When the dimension of the space is smaller than or equals to 7, the existence of weak solution is obtained by combining the Mountain Pass Lemma with some delicate estimate on the Talenti’s functions. When the dimension of the space is larger than or equals to 8, the above argument no longer works. By introducing an appropriate truncation on the nonlocal coefficient, it is shown that the problem admits a nontrivial solution under appropriate conditions on the parameter.

In this paper, we are concerned with the three-dimensional nonhomogeneous Bénard system with density-dependent viscosity in bounded domain. The global well-posedness of strong solution is established, provided that the initial total mass (|rho _{0}|_{L^{1}}) is suitably small. In particular, the initial velocity and temperature can be arbitrarily large. Moreover, the exponential decay of strong solution is also obtained. It is worth noting that the vacuum of initial density is allowed.

A functional Hilbert space is the Hilbert space ℋ of complex-valued functions on some set (Theta subseteq mathbb{C}) such that the evaluation functionals (varphi _{tau }left ( fright ) =fleft ( tau right ) ), (tau in Theta ), are continuous on ℋ. The Berezin number of an operator (X) is defined by (mathbf{ber}(X)=underset{tau in {Theta } }{sup }big vert widetilde{X}(tau )big vert = underset{tau in {Theta } }{sup }big vert langle Xhat{k}_{tau },hat{k}_{tau }rangle big vert ), where the operator (X) acts on the reproducing kernel Hilbert space ({mathscr{H}}={mathscr{H}(}Theta )) over some (non-empty) set (Theta ). In this paper, we introduce a new family involving means (Vert cdot Vert _{sigma _{t}}) between the Berezin radius and the Berezin norm. Among other results, it is shown that if (Xin {mathscr{L}}({mathscr{H}})) and (f), (g) are two non-negative continuous functions defined on ([0,infty )) such that (f(t)g(t) = t,,(tgeqslant 0)), then

and

where (sigma ) is a mean dominated by the arithmetic mean (nabla ).

This paper is devoted to a charge transfer model with cross-diffusion under Neumann boundary conditions. We investigate how the cross-diffusion could destabilize the stable periodic solutions bifurcating from the unique positive equilibrium point. By the implicit function theorem and Floquet theory, we obtain some conditions on the self-diffusion and cross-diffusion coefficients under which the stable periodic solutions can become unstable. New irregular Turing patterns then generate by the destabilization of stable spatially homogeneous periodic solutions. We also present some numerical simulations to further support the results of theoretical analysis.

In this paper, we study the variational inequality over the solution set of the split common solution problem with multiple output sets for monotone operator equations in real Hilbert spaces. We propose a new approach for solving this problem without depending on the norm of the transfer mappings. Some of its applications for solving the variational inequality over the solution set of the split common fixed point problem with multiple output sets and related problems are also mentioned. Finally, we give numerical examples to illustrate the performance of the proposed algorithm and compare it with several existing previous algorithms.

Motivated by the incompressible limit of a cell density model, we propose a free boundary tumor growth model where the pressure satisfies an obstacle problem on an evolving domain (Omega (t)), and the coincidence set (Lambda (t)) captures the emerging necrotic core. We contribute to the analytical characterization of the solution structure in the following two aspects. By deriving a semi-analytical solution and studying its dynamical behavior, we obtain quantitative transitional properties of the solution separating phases in the development of necrotic cores and establish its long time limit with the traveling wave solutions. Also, we prove the existence of traveling wave solutions incorporating non-zero outer densities outside the tumor bulk, provided that the size of the outer density is below a threshold.

Tumour growth is a complex process influenced by various factors, including cell proliferation, migration, and chemotaxis. In this study, a biphasic chemotaxis model for tumour growth is considered, and the effect of chemotherapy on the growth process is investigated. We use optimal control theory to derive the optimized treatment strategy that minimises the tumour size while minimising the toxicity associated with chemotherapy. Moreover, the existence, uniqueness, and strong solution estimates for the biphasic chemotaxis model subsystem in one dimension are derived. These results are achieved through semigroup theory and the truncation method. In addition, the research provides evidence of the existence of an optimal pair through the utilization of the minimising sequence technique. It also demonstrates the differentiability of the mapping from control variable to state variable and establishes the first-order necessary optimality condition. Lastly, a sequence of numerical simulations are presented to showcase the impact of chemotherapy and the influence of parameters in restraining tumour growth when applied in an optimized manner. Our results show that optimal control can provide a more effective and personalised treatment for cancer patients, and the approach can be extended to other tumour growth models.

This work focusses on the well–posedness and the exponential stability of a simplified porous elastic system. By omitting the second time derivative term of the function (varphi ) in the porous elastic system, we make the system free of the damaging consequences of the second spectrum. We consider an elastic dissipation and prove the well–posedness by the use of Hille–Yosida therorem and some arguments of elliptic equations. Next, we use the multiplier method and establish an exponential decay result without any restriction on coefficients of the system.