Manual Symmetry of C^1 solutions of p-Laplace equations in R^N

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In [ 1 ], Ding and Shen considered the following problems:. By constructing some auxiliary functions and using the differential inequality technique, they established the conditions on functions f , g , h , and u 0 to ensure that the solution u blows up at some time. In addition, an upper bound and a lower bound of the blow-up time were obtained. The method in [ 1 ] is not suitable for the study of problem 1. Zhang et al. By constructing some auxiliary functions and using parabolic maximum principles, they set up the conditions on functions a , f , g , h , and u 0 to guarantee that the solution either blows up in a finite time or exists globally.

Moreover, an upper estimate of the blow-up rate and an upper bound of the blow-up time are given. They also obtained an upper estimate of the global solution. We intend to use the methods in [ 16 ] and [ 22 ] to study problem 1. Since the principal parts of the two equations are different in problems 1.

Therefore, the key to our research is to construct some new auxiliary functions. By using these new auxiliary functions, parabolic maximum principles, and differential inequality techniques, we complete the study of problem 1. We proceed as follows. In Sect.

An upper estimate of the blow-up solution and an upper bound of the blow-up time are also given. At the same time, we also obtain an upper estimate of the global solution. We also adopt summation convection, for example,. In order to study the blow-up solution of 1. Let u be a nonnegative classical solution of 1. Assume that the following three assumptions are true :.

Symmetry of C1 Solutions of p-Laplace Equations in ℝN : Advanced Nonlinear Studies

It follows from 2. Inserting 2. Substituting 2. Assumption 2. Hence, we have.

The regularity assumptions on functions f , g , and h in Sect. First, we consider case a. With 2. Then, we consider case b. Now we have. Finally, we consider case c. Making use of the boundary condition of 1. Parabolic maximum principles, 2. In order to complete the study of the global solution to 1.


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Assume that the following three assumptions are satisfied :. By using the reasoning process 2. It follows from 3. The parabolic maximum principle guarantees that in the following three possible cases, Q may take its nonpositive minimum value:. First, case a is considered. Then, case b is considered.

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Repeating the reasoning process of 2. Finally, case c is considered. With the aid of the reasoning process in 2. Combining 3. In other words, we have. This shows that u must be a global solution. In this section, we give two examples to illustrate the results of Theorems 2. Let u be a nonnegative classical solution of the following problem:.

It is easy to see that. We easily verify that the three assumptions 2. It is easy to check that the three assumptions 3. In this paper, we research the blow-up and global solutions of p -Laplacian parabolic problem 1. We find that it is difficult to study the existence of blow-up and global solutions of problem 1. The main reason for this is that the boundary conditions in problems 1. As in [ 16 ] and [ 22 ], we combine the parabolic maximum principle with differential inequality to study problem 1.

The difficulty of using this method is the need to construct some appropriate auxiliary functions. Using these auxiliary functions, the parabolic maximum principle, and the differential inequality technique, we complete the study of 1. We set up the conditions on functions f , g , h , and u 0 to ensure that the solution of 1.

In addition, an upper estimate of the global solution and the blow-up rate are obtained. We also give an upper bound for the blow-up time. The author declares that there is no conflict of interests regarding the publication of this paper. Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. National Center for Biotechnology Information , U. Journal of Inequalities and Applications.

J Inequal Appl. Published online Apr 3. Juntang Ding. China Find articles by Juntang Ding. Author information Article notes Copyright and License information Disclaimer. Juntang Ding, Email: nc. Corresponding author.

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Received Dec 28; Accepted Mar Keywords: Blow-up, p -Laplacian equation, Nonlinear boundary condition. Introduction For more than ten years, many authors have discussed the blow-up phenomena of p -Laplacian parabolic problems. Also, this will satisfy each of the four original boundary conditions. In each of these cases the lone nonhomogeneous boundary condition will take the place of the initial condition in the heat equation problems that we solved a couple of sections ago.

We will apply separation of variables to each problem and find a product solution that will satisfy the differential equation and the three homogeneous boundary conditions.

Symmetry of C1 Solutions of p-Laplace Equations in ℝN

The process is nearly identical in many ways to what we did when we were solving the heat equation. Note that in this case, unlike the heat equation we must solve the boundary value problem first. Taking a change of letters into account the eigenvalues and eigenfunctions for the boundary value problem here are,. You should verify this by plugging this into the differential equation and checking that it is in fact a solution.

That is not really a problem however because we now have enough information to form the product solution for this partial differential equation. The Principle of Superposition then tells us that a solution to the partial differential equation is,. The difference here is that the coefficients of the Fourier sine series are now,.


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Now, at this point we need to choose a separation constant. Not only that but sometimes all it takes is a small change in the boundary conditions it force the change. So, the eigenvalues and eigenfunctions for the first boundary value problem are,. So, applying the boundary condition to this gives,. No matter what kind of boundary conditions we have they will work the same. This specifies the temperature on the boundary of the disk.