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In this paper we consider a free boundary problem modeling tumor growth with angiogenesis. The model is a free boundary problem of a system of partial differential equations. With Robin boundary, the model contains an ordinary differential equation describing the radius of tumor cell and two parabolic equations describing the evolution of nutrient concentration and inhibitor concentration, respectively. We study the number of the stationary solution of above problem, the existence and uniqueness of local solution and global solution, and the asymptotic behavior of the solution.

In this paper we study a granuloma model in visceral leishmaniasis, which contains eleven coupled reaction diffusion equations. The existence and uniqueness of the model in the local solution is proved by using the Banach Fixed Point Theorem and theory of parabolic equation. Then, the existence and uniqueness of the global solution are obtained by using the extension method.

This paper is devoted to the existence and Lipschitz continuity of global conservative weak solutions in time for the modified two-component Camassa–Holm system on the real line. We obtain the global weak solutions via a coordinate transformation into the Lagrangian coordinates. The key ingredients in our analysis are the energy density given by the positive Radon measure and the proposed new distance functions as well.

In this paper, we study the periodic Hunter–Saxton equation with weak dissipation. We first establish the local existence of strong solutions, blow-up scenario and blow-up criteria of the equation. Then, we investigate the blow-up rate for the blowing-up solutions to the equation. Finally, we prove that the equation has global solutions.

In this article, we study a mathematical model for glioma cells outside tumor spheroid core. It contains matrix metalloproteases and nutrient concentrations, and takes into account of the effects of chemotaxis, haptotaxis, cell–cell adhesion, proliferation and shedding. The model consists of three semi-linear parabolic partial differential equations and an ordinary differential equation. By using the Banach fixed point theorem, the parabolic L p -theory, the parabolic Schauder estimates and the extension method, we prove that this system has a unique global solution.

We consider the problem of finding limit cycles for a class of quintic polynomial differential systems and their global shape in the plane. An answer to this problem can be given using the averaging theory. More precisely, we analyze the global shape of the limit cycles which bifurcate from a Hopf bifurcation and periodic orbits of the linear center ẋ = −y, ẏ = x, respectively.

In this paper, we consider a periodic generalized Hunter–Saxton equation. We obtain the existence of global weak solutions to the equation. First, we give the well-posedness result of the viscous approximate equations and establish the basic energy estimates. Then, we show that the limit of the viscous approximation solutions is a global weak solution to the equation.

In this paper, we study the periodic Hunter–Saxton equation parametrized by the speed κ of the Galilean frame. First, we obtain the local existence and the blow-up scenario of strong solutions to the equation. Second, we establish two blow-up results. Third, we present the blow-up rate of strong solutions to the equation. Finally, we prove that the equation also has global solutions.

In this paper we study a mathematical model describing the growth of a solid in the presence of an immune system response. The model is strongly coupled degenerate reaction–diffusion system, in which the equations involve discontinuous terms. By using the approximation method combined with energy estimates and the bootstrap arguments, we prove that this system has a global classical solution.

In this paper we study the analyticity of solutions to a free boundary problem modeling the growth of multi-layer tumors. The problem consists of three elliptic equations defined on a strip-like domain in R n + 1 , with one part of the boundary moving and a priori unknown. The evolution of the moving boundary is governed by a Stefan type equation, with the surface tension effect taken into consideration. Due to the unknown boundary and surface tension effect, this problem is an essentially nonlinear problem. By following a functional analytical approach and the theory of maximal regularity, we prove that solutions of this free boundary problem are real analytic in time and space for any positive time, even if the given initial data admit mild regularity only.

We study the solvability in anisotropic Sobolev spaces for the Cauchy problem of the nonisotropically perturbed sixth-order nonlinear Schrodinger equation: , where a, b are real constants, 1 ≤ d < n and x = (x 1, x 2, …, x n ) ∈ R n , t ∈ R and f(u) is a given complex function. By using Kato's method, we establish local well-posedness for the equation. We also obtain its global solvability by the anisotropic Sobolev embedding relations and the energy method.

In this paper we study a mathematical model of antiangiogenesis in tumor growth. The model consists of three semi-linear parabolic PDEs. By applying the Banach fixed point method, the parabolic Schauder estimates, parabolic L p estimates and the continuation method, we prove that this system has a unique global classical solution.

In this paper we study a free boundary problem modelling of the use of macrophages as vehicles for drug delivery to hypoxic tumor sites. The model consists of five semi-linear parabolic partial differential equations. By applying the parabolic Schauder estimates and the Banach fixed point method, we prove that this system has a unique local classical solution. Using this result, parabolic L p estimates, the monotone method in the theory of reaction diffusion equations and the extension method, we further prove that if the tumor does not vanish at finite time then the solution exists for all t ⩾ 0 .

In this paper we study global existence of solutions of a mathematical model for drug transport in tumor multicell spheroids. The model is a free boundary problem of a system of partial differential equations. It contains one nonlinear first-order equation describing the distribution of live tumor cells, and two nonlinear reaction diffusion equations describing the evolution of nutrient concentration and drug concentration, respectively. By using the method of characteristics for first-order equations, the L^p-theory for parabolic equations, the Banach fixed point theorem and the extension method, we prove that this problem has a unique global solution.

In this paper, we consider a periodic Hunter–Saxton equation with weak dissipation. We obtain the existence of global weak solutions to the equation for any initial data in H1(𝕊).

In this paper we study the Cauchy problem of the non-isotropically perturbed fourth-order nonlinear Schrodinger type equation: i u t + △ u + a ∑ i = 1 d u x i x i x i x i + g ( x , | u | ) u = 0 ( ( x 1 , x 2 , … , x n ) ∈ R n , t ⩾ 0 ), where a is a real constant, 1 ⩽ d n is an integer, g ( x , | u | ) u is a nonlinear function which behaves like | u | α u for some constant α > 0 . By using Kato method, we prove that this perturbed fourth-order Schrodinger type equation is locally well-posed with initial data belonging to the non-isotropic Sobolev spaces H x → s 1 H y → s 2 provided that s 1 , s 2 satisfy the conditions: s 1 ⩾ 0 , s 2 ⩾ 0 for 0 α 8 2 n − d or 2 s 1 + 4 s 2 > 2 n − d − 8 α for α ⩾ 8 2 n − d with some additional conditions. Furthermore, by using non-isotropic Sobolev inequality and energy method, we obtain some global well-posedness results for initial data belonging to non-isotropic Sobolev spaces H x → 2 H y → 1 .