This paper is concerned with the asymptotic behavior of solutions of non-autonomous reaction-diffusion equation with delays. The well-posedness theory of equation for the initial data belonging to $ C_{L^{r}(\Omega)}(1<r<\infty)$ and $ C_{W^{1, r}(\Omega)}(1<r<N)$ is established respectively. In addition, the existence of uniform attractors in $ C_{L^{r}(\Omega)}$ for the family of processes with translation bounded external force is proved. Moreover, the long time behavior of solution with higher regularity in $ C_{W^{1, r}(\Omega)}$ is considered as well.

We consider periodic solutions of the following nonlinear system associated with the fractional Laplacian

where F : $ \mathbb{R}^{2} \rightarrow \mathbb{R}$ is a smooth double-well potential. For the case that F is even in its two variables we obtain the existence of more and more periodic solutions with large period, by using Clark’s theorem. For the case that F is only even in its the second variable and the origin is a saddle critical point of F, we give two periodic solutions by using Morse theory.

In this paper, we study the existence of at least two, three and infinitely many solutions for nonlinear problems on the Sierpiński gasket, modelling some physical phenomena such as reaction-diffusion problems, elastic properties of fractal media and flow through fractal non-smooth domains. We will obtain the existence of two weak solutions when nonlinear term f(x,t) is non-negative, when it is non-positive in neighbored of zero and otherwise is positive we will show the existence of three weak solutions, and when it is odd we will get the existence of infinitely many solutions. The results are proved by using some critical point theorems.

In this paper, we consider the following critical Choquard equation

where µ>0 is a parameter, ν∈(0,3),p∈(4,6) and f,g are continuous functions. For µ small enough, by using Lusternik-Schnirelmann category theory, we establish a relationship between the number of solutions and the category of the global maximum set of g.

In this paper, we obtain a parameter type logarithmic Sobolev inequality with weight and a modified parameter type logarithmic Sobolev inequality with weight on Euclidean space based on the parameter type logarithmic Caffarelli-Kohn-Nirenberg inequality on Euclidean space, respectively. By virtue of the convexity of a special function equivalent to a logarithmic Hölder inequality, and combining the Sobolev in-equality and Gagliardo-Nirenberg inequality on Hörmander’s vector fields, we also derive a logarithmic Sobolev inequality and a parameter type logarithmic Gagliardo-Nirenberg inequality on Hörmander’s vector fields, respectively. In addition, a parameter type logarithmic Sobolev inequality on Hörmander’s vector fields is given by suitable stretching transformation.

In this work, we study the existence of one-sign solutions for the following problem:

where N≥3, λ is a real parameter and $ a \in C_{\mathrm{loc}}^{\alpha}\left(\mathbb{R}^{N}, \mathbb{R}\right)$ for some α∈(0,1) is a weighted function, $ f: \mathbb{R} \rightarrow \mathbb{R}$ is a Hölder continuous function with exponent αα such that f(s)s>0 for any s≠0. We determine the intervals of λ for the existence, exact multiplicity of one-sign solutions for this problem. We use bifurcation techniques and the approximation of connected components to prove our main results.

In this article, we investigate the well-posedness of the initial-boundary value problem (I-B-V problem) for the fifth-order KdV equation posed on a finite domain with nonlinear boundary conditions. Firstly, we establish various a priori estimates, including Kato smoothing effects, sharp trace regularity, and nonlinear estimates. Subsequently, we demonstrate that the initial-boundary value problem of the fifth-order KdV equation with quadratic boundary feedbacks is locally well-posed for the appropriately chosen initial value and boundary values.