We establish the nonexistence of nontrivial ancient solutions to the nonlinear heat equation ut=Δu+|u|p−1u which are smaller in absolute value than the self-similar radial singular steady state, provided that the exponent p is strictly between Serrin's exponent and that of Joseph and Lundgren. This result was previously established by Fila and Yanagida [Tohoku Math. J. (2011)] by using forward self-similar solutions as barriers. In contrast, we apply a sweeping argument with a family of time independent weak supersolutions. Our approach naturally lends itself to yield an analogous Liouville type result for the steady state problem in higher dimensions. In fact, in the case of the critical Sobolev exponent we show the validity of our results for solutions that are smaller in absolute value than a 'Delaunay'-type singular solution.
Citation: Christos Sourdis. A Liouville theorem for ancient solutions to a semilinear heat equation and its elliptic counterpart[J]. Electronic Research Archive, 2021, 29(5): 2829-2839. doi: 10.3934/era.2021016
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We establish the nonexistence of nontrivial ancient solutions to the nonlinear heat equation ut=Δu+|u|p−1u which are smaller in absolute value than the self-similar radial singular steady state, provided that the exponent p is strictly between Serrin's exponent and that of Joseph and Lundgren. This result was previously established by Fila and Yanagida [Tohoku Math. J. (2011)] by using forward self-similar solutions as barriers. In contrast, we apply a sweeping argument with a family of time independent weak supersolutions. Our approach naturally lends itself to yield an analogous Liouville type result for the steady state problem in higher dimensions. In fact, in the case of the critical Sobolev exponent we show the validity of our results for solutions that are smaller in absolute value than a 'Delaunay'-type singular solution.
Multi-nominal data are common in scientific and engineering research such as biomedical research, customer behavior analysis, network analysis, search engine marketing optimization, web mining etc. When the response variable has more than two levels, the principle of mode-based or distribution-based proportional prediction can be used to construct nonparametric nominal association measure. For example, Goodman and Kruskal [3,4] and others proposed some local-to-global association measures towards optimal predictions. Both Monte Carlo and discrete Markov chain methods are conceptually based on the proportional associations. The association matrix, association vector and association measure were proposed by the thought of proportional associations in [9]. If there is no ordering to the response variable's categories, or the ordering is not of interest, they will be regarded as nominal in the proportional prediction model and the other association statistics.
But in reality, different categories in the same response variable often are of different values, sometimes much different. When selecting a model or selecting explanatory variables, we want to choose the ones that can enhance the total revenue, not just the accuracy rate. Similarly, when the explanatory variables with cost weight vector, they should be considered in the model too. The association measure in [9],
To implement the previous adjustments, we need the following assumptions:
It needs to be addressed that the second assumption is probably not always the case. The law of large number suggests that the larger the sample size is, the closer the expected value of a distribution is to the real value. The study of this subject has been conducted for hundreds of years including how large the sample size is enough to simulate the real distribution. Yet it is not the major subject of this article. The purpose of this assumption is nothing but a simplification to a more complicated discussion.
The article is organized as follows. Section 2 discusses the adjustment to the association measure when the response variable has a revenue weight; section 3 considers the case where both the explanatory and the response variable have weights; how the adjusted measure changes the existing feature selection framework is presented in section 4. Conclusion and future works will be briefly discussed in the last section.
Let's first recall the association matrix
γs,t(Y|X)=E(p(Y=s|X)p(Y=t|X))p(Y=s)=α∑i=1p(X=i|Y=s)p(Y=t|X=i);s,t=1,2,..,βτY|X=ωY|X−Ep(Y)1−Ep(Y)ωY|X=EX(EY(p(Y|X)))=β∑s=1α∑i=1p(Y=s|X=i)2p(X=i)=β∑s=1γssp(Y=s) | (1) |
Our discussion begins with only one response variable with revenue weight and one explanatory variable without cost weight. Let
Definition 2.1.
ˆωY|X=β∑s=1α∑i=1p(Y=s|X=i)2rsp(X=i)=β∑s=1γssp(Y=s)rsrs>0,s=1,2,3...,β | (2) |
Please note that
It is easy to see that
Example.Consider a simulated data motivated by a real situation. Suppose that variable
1000 | 100 | 500 | 400 | 500 | 300 | 200 | 1500 | |||
200 | 1500 | 500 | 300 | 500 | 400 | 400 | 50 | |||
400 | 50 | 500 | 500 | 500 | 500 | 300 | 700 | |||
300 | 700 | 500 | 400 | 500 | 400 | 1000 | 100 | |||
200 | 500 | 400 | 200 | 200 | 400 | 500 | 200 |
Let us first consider the association matrix
0.34 | 0.18 | 0.27 | 0.22 | 0.26 | 0.22 | 0.27 | 0.25 | |||
0.13 | 0.48 | 0.24 | 0.15 | 0.25 | 0.24 | 0.29 | 0.23 | |||
0.24 | 0.28 | 0.27 | 0.21 | 0.25 | 0.24 | 0.36 | 0.15 | |||
0.25 | 0.25 | 0.28 | 0.22 | 0.22 | 0.18 | 0.14 | 0.46 |
Please note that
The correct prediction contingency tables of
471 | 6 | 121 | 83 | 98 | 34 | 19 | 926 | |||
101 | 746 | 159 | 107 | 177 | 114 | 113 | 1 | |||
130 | 1 | 167 | 157 | 114 | 124 | 42 | 256 | |||
44 | 243 | 145 | 85 | 109 | 81 | 489 | 6 | |||
21 | 210 | 114 | 32 | 36 | 119 | 206 | 28 |
The total number of the correct predictions by
total revenue | average revenue | |||
0.3406 | 0.456 | 4313 | 0.4714 | |
0.3391 | 0.564 | 5178 | 0.5659 |
Given that
In summary, it is possible for an explanatory variable
Let us further discuss the case with cost weight vector in predictors in addition to the revenue weight vector in the dependent variable. The goal is to find a predictor with bigger profit in total. We hence define the new association measure as in 3.
Definition 3.1.
ˉωY|X=α∑i=1β∑s=1p(Y=s|X=i)2rscip(X=i) | (3) |
Example. We first continue the example in the previous section with new cost weight vectors for
total profit | average profit | ||||
0.3406 | 0.3406 | 1.3057 | 12016.17 | 1.3132 | |
0.3391 | 0.3391 | 1.8546 | 17072.17 | 1.8658 |
By
We then investigate how the change of cost weight affect the result. Suppose the new weight vectors are:
total profit | average profit | ||||
0.3406 | 0.3406 | 1.7420 | 15938.17 | 1.7419 | |
0.3391 | 0.3391 | 1.3424 | 12268.17 | 1.3408 |
Hence
By the updated association defined in the previous section, we present the feature selection result in this section to a given data set
At first, consider a synthetic data set simulating the contribution factors to the sales of certain commodity. In general, lots of factors could contribute differently to the commodity sales: age, career, time, income, personal preference, credit, etc. Each factor could have different cost vectors, each class in a variable could have different cost as well. For example, collecting income information might be more difficult than to know the customer's career; determining a dinner waitress' purchase preference is easier than that of a high income lawyer. Therefore we just assume that there are four potential predictors,
total profit | average profit | ||||
7 | 0.3906 | 3.5381 | 35390 | 3.5390 | |
4 | 0.3882 | 3.8433 | 38771 | 3.8771 | |
4 | 0.3250 | 4.8986 | 48678 | 4.8678 | |
8 | 0.3274 | 3.7050 | 36889 | 3.6889 |
The first variable to be selected is
total profit | average profit | ||||
28 | 0.4367 | 1.8682 | 18971 | 1.8971 | |
28 | 0.4025 | 2.1106 | 20746 | 2.0746 | |
56 | 0.4055 | 1.8055 | 17915 | 1.7915 | |
16 | 0.4055 | 2.3585 | 24404 | 2.4404 | |
32 | 0.3385 | 2.0145 | 19903 | 1.9903 |
As we can see, all
In summary, the updated association with cost and revenue vector not only changes the feature selection result by different profit expectations, it also reflects a practical reality that collecting information for more variables costs more thus reduces the overall profit, meaning more variables is not necessarily better on a Return-Over-Invest basis.
We propose a new metrics,
The presented framework can also be applied to high dimensional cases as in national survey, misclassification costs, association matrix and association vector [9]. It should be more helpful to identify the predictors' quality with various response variables.
Given the distinct character of this new statistics, we believe it brings us more opportunities to further studies of finding the better decision for categorical data. We are currently investigating the asymptotic properties of the proposed measures and it also can be extended to symmetrical situation. Of course, the synthetical nature of the experiments in this article brings also the question of how it affects a real data set/application. It is also arguable that the improvements introduced by the new measures probably come from the randomness. Thus we can use
[1] |
Monotonicity for elliptic equations in an unbounded Lipschitz domain. Comm. Pure Appl. Math. (1997) 50: 1089-1111. ![]() |
[2] |
Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Comm. Pure Appl. Math. (1989) 42: 271-297. ![]() |
[3] |
Existence of positive weak solutions with a prescribed singular set of semilinear elliptic equations. J. Geom. Anal. (1999) 9: 221-246. ![]() |
[4] |
(2011) Stable Solutions of Elliptic Partial Differential Equations. Boca Raton: Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math. 143, CRC Press. ![]() |
[5] |
On the classification of solutions of the Lane-Emden equation on unbouned domains of RN. J. Math. Pures Appl. (2007) 87: 537-561. ![]() |
[6] |
Homoclinic and heteroclinic orbits for a semilinear parabolic equation. Tohoku Math. J. (2011) 63: 561-579. ![]() |
[7] |
Global and local behavior of positive solutions of nonlinear elliptic equations. Comm. Pure Appl. Math. (1981) 34: 525-598. ![]() |
[8] |
On the stability and instability of positive steady states of a semilinear heat equation in Rn. Comm. Pure Appl. Math. (1992) 45: 1153-1181. ![]() |
[9] |
Quasilinear Dirichlet problems driven by positive sources. Arch. Rational Mech. Anal. (1973) 49: 241-269. ![]() |
[10] |
G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302
![]() |
[11] | Strong comparison theorems for elliptic equations of second order. J. Math. Mech. (1961) 10: 431-440. |
[12] |
An ODE approach to the multiplicity of self-similar solutions for semi-linear heat equations. Proc. Roy. Soc. Edinburgh Sect. A (2006) 136: 807-835. ![]() |
[13] |
Entire and ancient solutions of a supercritical semilinear heat equation. Discrete Cont. Dynamical Syst. (2021) 41: 413-438. ![]() |
[14] |
Singularity and decay estimates in superlinear problems via Liouville-type theorems. Part I: Elliptic equations and systems. Duke Math. J. (2007) 139: 555-579. ![]() |
[15] |
Singularity and decay estimates in superlinear problems via Liouville-type theorems. Part II: parabolic equations. Indiana Univ. Math. J. (2007) 56: 879-908. ![]() |
[16] |
A Liouville property and quasiconvergence for a semilinear heat equation. J. Differential Equations (2005) 208: 194-214. ![]() |
[17] |
Liouville theorems for scaling invariant superlinear parabolic problems with gradient structure. Math. Ann. (2016) 364: 269-292. ![]() |
[18] | P. Quittner, Optimal Liouville theorems for superlinear parabolic problems, Duke Math. J., to appear. |
[19] |
P. Quittner and Ph. Souplet, Superlinear Parabolic Problems. Blow-Up, Global Existence and Steady States, 2nd edition, Birkhäuser Advanced Texts, Birkhäuser, Basel, 2019. doi: 10.1007/978-3-030-18222-9
![]() |
[20] |
R. M. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, in Topics in Calculus of Variations, Lecture Notes in Math. Springer, 1365 (1989), 120–154. doi: 10.1007/BFb0089180
![]() |
[21] | C. Sourdis, A Liouville property for eternal solutions to a supercritical semilinear heat equation, preprint, arXiv: 1909.00498. |
1000 | 100 | 500 | 400 | 500 | 300 | 200 | 1500 | |||
200 | 1500 | 500 | 300 | 500 | 400 | 400 | 50 | |||
400 | 50 | 500 | 500 | 500 | 500 | 300 | 700 | |||
300 | 700 | 500 | 400 | 500 | 400 | 1000 | 100 | |||
200 | 500 | 400 | 200 | 200 | 400 | 500 | 200 |
0.34 | 0.18 | 0.27 | 0.22 | 0.26 | 0.22 | 0.27 | 0.25 | |||
0.13 | 0.48 | 0.24 | 0.15 | 0.25 | 0.24 | 0.29 | 0.23 | |||
0.24 | 0.28 | 0.27 | 0.21 | 0.25 | 0.24 | 0.36 | 0.15 | |||
0.25 | 0.25 | 0.28 | 0.22 | 0.22 | 0.18 | 0.14 | 0.46 |
471 | 6 | 121 | 83 | 98 | 34 | 19 | 926 | |||
101 | 746 | 159 | 107 | 177 | 114 | 113 | 1 | |||
130 | 1 | 167 | 157 | 114 | 124 | 42 | 256 | |||
44 | 243 | 145 | 85 | 109 | 81 | 489 | 6 | |||
21 | 210 | 114 | 32 | 36 | 119 | 206 | 28 |
total revenue | average revenue | |||
0.3406 | 0.456 | 4313 | 0.4714 | |
0.3391 | 0.564 | 5178 | 0.5659 |
total profit | average profit | ||||
0.3406 | 0.3406 | 1.3057 | 12016.17 | 1.3132 | |
0.3391 | 0.3391 | 1.8546 | 17072.17 | 1.8658 |
total profit | average profit | ||||
0.3406 | 0.3406 | 1.7420 | 15938.17 | 1.7419 | |
0.3391 | 0.3391 | 1.3424 | 12268.17 | 1.3408 |
total profit | average profit | ||||
7 | 0.3906 | 3.5381 | 35390 | 3.5390 | |
4 | 0.3882 | 3.8433 | 38771 | 3.8771 | |
4 | 0.3250 | 4.8986 | 48678 | 4.8678 | |
8 | 0.3274 | 3.7050 | 36889 | 3.6889 |
total profit | average profit | ||||
28 | 0.4367 | 1.8682 | 18971 | 1.8971 | |
28 | 0.4025 | 2.1106 | 20746 | 2.0746 | |
56 | 0.4055 | 1.8055 | 17915 | 1.7915 | |
16 | 0.4055 | 2.3585 | 24404 | 2.4404 | |
32 | 0.3385 | 2.0145 | 19903 | 1.9903 |
1000 | 100 | 500 | 400 | 500 | 300 | 200 | 1500 | |||
200 | 1500 | 500 | 300 | 500 | 400 | 400 | 50 | |||
400 | 50 | 500 | 500 | 500 | 500 | 300 | 700 | |||
300 | 700 | 500 | 400 | 500 | 400 | 1000 | 100 | |||
200 | 500 | 400 | 200 | 200 | 400 | 500 | 200 |
0.34 | 0.18 | 0.27 | 0.22 | 0.26 | 0.22 | 0.27 | 0.25 | |||
0.13 | 0.48 | 0.24 | 0.15 | 0.25 | 0.24 | 0.29 | 0.23 | |||
0.24 | 0.28 | 0.27 | 0.21 | 0.25 | 0.24 | 0.36 | 0.15 | |||
0.25 | 0.25 | 0.28 | 0.22 | 0.22 | 0.18 | 0.14 | 0.46 |
471 | 6 | 121 | 83 | 98 | 34 | 19 | 926 | |||
101 | 746 | 159 | 107 | 177 | 114 | 113 | 1 | |||
130 | 1 | 167 | 157 | 114 | 124 | 42 | 256 | |||
44 | 243 | 145 | 85 | 109 | 81 | 489 | 6 | |||
21 | 210 | 114 | 32 | 36 | 119 | 206 | 28 |
total revenue | average revenue | |||
0.3406 | 0.456 | 4313 | 0.4714 | |
0.3391 | 0.564 | 5178 | 0.5659 |
total profit | average profit | ||||
0.3406 | 0.3406 | 1.3057 | 12016.17 | 1.3132 | |
0.3391 | 0.3391 | 1.8546 | 17072.17 | 1.8658 |
total profit | average profit | ||||
0.3406 | 0.3406 | 1.7420 | 15938.17 | 1.7419 | |
0.3391 | 0.3391 | 1.3424 | 12268.17 | 1.3408 |
total profit | average profit | ||||
7 | 0.3906 | 3.5381 | 35390 | 3.5390 | |
4 | 0.3882 | 3.8433 | 38771 | 3.8771 | |
4 | 0.3250 | 4.8986 | 48678 | 4.8678 | |
8 | 0.3274 | 3.7050 | 36889 | 3.6889 |
total profit | average profit | ||||
28 | 0.4367 | 1.8682 | 18971 | 1.8971 | |
28 | 0.4025 | 2.1106 | 20746 | 2.0746 | |
56 | 0.4055 | 1.8055 | 17915 | 1.7915 | |
16 | 0.4055 | 2.3585 | 24404 | 2.4404 | |
32 | 0.3385 | 2.0145 | 19903 | 1.9903 |