In this paper, inequality techniques, stochastic analysis and algebraic methods are used to analyze the robustness of the stability of recurrent neural networks containing Takagi-Sugeno fuzzy rules. By solving the transcendental equations, the upper bounds of time delay and noise intensity are given, and the dynamic relationship between the two disturbance factors is derived. Finally, numerical examples are given to verify the results of this paper.
Citation: Wenxiang Fang, Tao Xie. Robustness analysis of stability of Takagi-Sugeno type fuzzy neural network[J]. AIMS Mathematics, 2023, 8(12): 31118-31140. doi: 10.3934/math.20231593
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In this paper, inequality techniques, stochastic analysis and algebraic methods are used to analyze the robustness of the stability of recurrent neural networks containing Takagi-Sugeno fuzzy rules. By solving the transcendental equations, the upper bounds of time delay and noise intensity are given, and the dynamic relationship between the two disturbance factors is derived. Finally, numerical examples are given to verify the results of this paper.
Let Ω={x∈¯RN+:|x|≥1}, RN+={x=(x1,x2,⋯,xN)∈RN:xN>0} and N≥2. The boundary of Ω is denoted by
∂Ω=1⋃i=0Γi, |
where Γ0={x∈Ω:xN=0} and Γ1={x∈Ω:xN>0,|x|=1}. We are concerned with the existence an nonexistence of weak solutions to the evolution inequality
∂ku∂tk−Δ(|u|m−1u)+μ|x|2x⋅∇(|u|m−1u)≥|x|a|u|pin (0,∞)×Ω, | (1.1) |
where u=u(t,x), k≥1 is an integer, p>m≥1, μ,a∈R and ⋅ is the inner product in RN. Problem (1.1) is considered under the Dirichlet-type boundary conditions
{u≥0on (0,∞)×Γ0,|u|m−1u≥fon (0,∞)×Γ1, | (1.2) |
where f=f(x).
The issue of existence and nonexistence of solutions to higher order (in time) evolution inequalities has been studied in several papers. For instance, Hamidi and Laptev [1] investigated the nonexistence of weak solutions to higher-order evolution inequalities of the form
{∂ku∂tk−Δu+λ|x|2u≥|u|pin (0,∞)×RN,∂k−1u∂tk−1(0,x)≥0in RN, | (1.3) |
where N≥3, λ≥−(N−22)2 and p>1. Namely, it was shown that, if one of the following assumptions is satisfied:
λ≥0,1<p≤1+22k+s∗; |
or
−(N−22)2≤λ<0,1<p≤1+22k−s∗, |
where
s∗=N−22+√λ+(N−22)2,s∗=s∗+2−N, |
then (1.3) admits no nontrivial weak solution. In [2], Caristi considered evolution inequalities of the form
∂ku∂tk−|x|σΔmu≥|u|qin (0,∞)×RN, | (1.4) |
where m is a positive integer, q>1 and σ≤2m. In the case σ=2m (critical degeneracy case) it was proven that, if k≥2, ∂k−1u∂tk−1(0,⋅)|x|−N∈L1(RN) with a positive average, and one of the following conditions holds:
(ⅰ) N≠2(j+1) for j=0,m−1 and 1<q≤k+1;
(ⅱ) N=2(j+1) with j=0,⋯,m−1 and q>1,
then (1.4) has no weak solution. In the case σ<2m (the subcritical degeneracy case), it was shown that, if k≥2, ∂ju∂tj(0,⋅)|x|−σ∈L1loc(RN) for j=0,k−2, ∂k−1u∂tk−1(0,⋅)|x|−σ∈L1(RN) with a positive average, and
q(k(N−2m)+2m−σ)≤Nk+2m−σ(k+1), |
then (1.4) has no weak solution. Very recently, Filippucci and Ghergu [3] investigated evolution inequalities of the form
∂ku∂tk+(−Δ)mu≥(K∗|u|p)|u|q,in (0,∞)×RN, | (1.5) |
where N,k,m≥1 are integers, p,q>0 and K∈C(R+,R+) satisfies: K(|x|)∈L1loc(RN) and inf0<r<RK(r)=K(R) for sufficiently large R. Namely, the authors proved the following results:
(ⅰ) If k is an even integer and q≥1, then (1.5) admits some positive solutions u∈C∞((0,∞)×RN) which verify ∂k−1u∂tk−1(0,⋅)<0 in RN;
(ⅱ) If p+q>2 and
lim supR→∞K(R)R2N+2mkp+q−N+2m(1−1k)>0, |
then (1.5) has no nontrivial solutions such that
∂k−1u∂tk−1≥0;or∂k−1u∂tk−1(0,⋅)∈L1(RN),∫RN∂k−1u∂tk−1(0,x)dx>0. |
Other contributions related to higher order (in time) evolution equations and inequalities can be found in [4,5,6,7].
In [8], Zheng and Wang studied the large time behavior of nonnegative solutions to parabolic equations of the form
∂u∂t−Δum−λx|x|2⋅∇um=|x|σup(u≥0)in (0,∞)×RN∖¯ω, | (1.6) |
where k∈R, σ>−2, p>m≥1 and ω is a bounded domain in RN containing the origin with a smooth boundary ∂ω. Problem (1.6) was investigated under the homogeneous Neumann boundary condition
∂um∂ν(t,x)=0on (0,∞)×∂ω | (1.7) |
and the homogeneous Dirichlet boundary condition
u(t,x)=0on (0,∞)×∂ω. | (1.8) |
For problem (1.6) under the boundary condition (1.7), it was shown that (under a certain regularity on the geometry of ω)
p∗={m+σ+2N+λ,ifλ>−N,∞,ifλ≤−N | (1.9) |
is critical in the sense of Fujita. When λ>2−N, it was proven that problem (1.6) under the boundary condition (1.8), admits the same Fujita critical exponent p∗. Other contributions related to parabolic equations involving terms of the form b(x)⋅∇um can be found in [9,10,11,12,13] (see also the references therein). Notice that in all the above mentioned references, only the parabolic case has been treated. Moreover, the considered solutions have been assumed to be positive. Very recently, in [14], the authors considered evolution inequalities of the form
∂ku∂tk−Δ(|u|m−1u)−λx|x|2⋅∇(|u|m−1u)≥|x|σ|u|pin (0,∞)×Bc1, | (1.10) |
under different types of boundary conditions, where p>m≥1, B1 denotes the open ball of radius 1 centered at the origin point in RN with N≥2 and Bc1 denotes the complement of B1. For instance, under the Dirichlet-type boundary condition
|u(t,x)|m−1u(t,x)≥f(x)on (0,∞)×∂B1, |
where f∈L1(∂B1) has a positive average, the authors proved that when σ>−2, (1.10) admits as Fujita critical exponent
pcr={m+m(σ+2)λ+N−2ifλ>2−N,∞ifλ≤2−N. |
More precisely, the authors proved the following results:
(ⅰ) If λ≤2−N and f∈L1(∂B1) has a positive average, then for all p>m, (1.10) admits no weak solution;
(ⅱ) If λ>2−N and f∈L1(∂B1) has a positive average, then for all m<p≤pcr, (1.10) admits no weak solution;
(ⅲ) If λ>2−N and p>pcr, then (1.10) admits (stationary) solutions for some f>0.
For more contributions related to the issue of existence and nonexistence of solutions to evolution equations and inequalities in exterior domains, see e.g., [15,16,17,18,19].
Our aim in this paper is to study the influence of the obstacle domain on the critical behavior of (1.10) by considering the half-unit ball instead of the unit ball. Before presenting our main results, we need to define weak solutions to (1.1) and (1.2).
Let
D=(0,∞)×Ωand∂Di=(0,∞)×Γi,i=0,1. |
Notice that ∂Di⊂D for all i=0,1. We introduce the functional space V defined as follows.
Definition 1.1. A function φ=φ(t,x) belongs to V, if the following conditions are satisfied:
(ⅰ) φ∈Ck,2t,x(D), φ≥0;
(ⅱ) supp(φ)⊂⊂D;
(ⅲ) φ=0 on ∂Di, i=0,1;
(ⅳ) ∂φ∂νi≤0 on ∂Di, i=0,1, where νi denotes the outward unit normal vector on Γi, relative to D.
Using standard integrations by parts, we define weak solutions to (1.1) and (1.2) as follows.
Definition 1.2. We say that u∈Lploc(D) is a weak solutions to (1.1) and (1.2), if
∫D|x|a|u|pφdxdt−∫∂D1∂φ∂ν1f(x)dσdt≤(−1)k∫Du∂kφ∂tkdxdt−∫D|u|m−1u(Δφ+μdiv(φx|x|2))dxdt | (1.11) |
for every φ∈V.
For μ∈R, let us introduce the parameter
αμ=−N+μ2+√μ+(N−μ2)2. | (1.12) |
Our main results are stated in the following theorem.
Theorem 1.3. Let N≥2, k≥1 (an integer) and μ,a∈R.
(I) Let f∈L1(Γ1) be such that
∫Γ1f(x)xNdσ>0. | (1.13) |
Assume that
p>m,(αμ+N−1)p−m(αμ+1+a+N)≤0. | (1.14) |
Then (1.1) and (1.2) admits no weak solution.
(II) If
p>m,(αμ+N−1)p−m(αμ+1+a+N)>0, | (1.15) |
then (1.1) and (1.2) admits (stationary) solutions in the sense of Definition 1.2, for some f>0.
The proof of part (Ⅰ) of Theorem 1.3 is based on nonlinear capacity estimates specifically adapted to the domain, the operator −Δ+μ|x|2x⋅∇ and the boundary conditions (1.2). Part (Ⅱ) is established by the construction of expilicit solutions.
Remark 1.4. (ⅰ) Let us point out that the used method in [8] for proving the blow-up of solutions to (1.6) requires the positivity of u. Namely, the authors used that functions of the form
ℓ↦wℓ(t):=∫RN∖ωu(t,x)ψl(|x|)dx |
are nondecreasing for sufficiently large ℓ, where ψℓ≥0 is a certain cut-off function. In this paper, no restriction on the sign of solutions is imposed. Moreover, even in the case of positive solutions, it is difficult to use the method in [8] for proving the blow-up of solutions in the hyperbolic case. Namely, in order to show the blow-up of solutions to (1.6), the authors proved that the function wℓ defined above, satisfies the differential inequality
dwℓdt≥γwpℓ, |
for a certain constant γ>0. A such inequality is related essentially to the parabolic nature of the problem.
(ⅱ) The emphasis of this paper is on blow up results. The existence result provided by part (Ⅱ) of Theorem 1.3 is a consequence of elliptic results. We refer to [20,21], where some regularization methods to deal with the degeneracy were used to obtain the strong solution with latent singularity. We refer also to [22,23], where global solutions have been obtained following the standard gradient flow method. It will be interested to see if such methods can be adapted to the case of problem (1.1).
(ⅲ) It is not difficult to show that for all μ∈R, one has
αμ+N−1>0. |
Hence, (1.14) is equivalent to
m<p≤m+m(a+2)αμ+N−1,a>−2. |
(ⅳ) From the above remark, we observe that (1.15) is equivalent to
a≤−2;orp>m+m(a+2)αμ+N−1,a>−2. |
Remark 1.5. (ⅰ) From Remark 1.4, we deduce that, if a≤−2, then (1.1) and (1.2) admits no critical behavior. However, if a>−2, then (1.1) and (1.2) admits as Fujita critical exponent the real number
p∗=p∗(m,a,μ,N)=m+m(a+2)αμ+N−1. |
(ⅱ) It is interesting to observe that p∗ is independent on k. This implies that Theorem 1.3 holds true in the parabolic (k=1) as well as hyperbolic (k=2) case.
Clearly, Theorem 1.3 yields existence and nonexistence results for the corresponding stationary problem
−Δ(|u|m−1u)+μ|x|2x⋅∇(|u|m−1u)≥|x|a|u|pin Ω | (1.16) |
under the Dirichlet-type boundary conditions
{u≥0on Γ0,|u|m−1u≥fon Γ1. | (1.17) |
Corollary 1.6. Let N≥2 and μ,a∈R.
(I) Let f∈L1(Γ1) be such that (1.13) holds. If (1.14) is satisfied, then (1.16) and (1.17) admits no weak solution.
(II) If (1.15) holds, then (1.16) and (1.17) admits solutions for some f>0.
The rest of the paper is organized as follows. In Section 2, we establish some preliminary lemmas that will be useful in the proof of our main results. Namely, we first prove an a priori estimate for problems (1.1) and (1.2). Next, we construct two families of functions belonging to V. The first family will be used in the proof of part (Ⅰ) of Theorem 1.3 in the case (αμ+N−1)p−m(αμ+1+a+N)<0, and the second family will be used in the proof of the critical case (αμ+N−1)p−m(αμ+1+a+N)=0. Finally, Section 3 is devoted to the proof of Theorem 1.3.
Throughout this paper, the letters C,Ci denote always generic positive constants whose values are unimportant and may vary at different occurrences.
Let N≥2, k≥1 (an integer), p>m≥1, μ,a∈R and f∈L1(Γ1). We denote by Lμ the differential operator given by
Lμϕ=Δϕ+μdiv(ϕx|x|2). |
For φ∈V, we introduce the integral terms
ω1(φ)=∫supp(φ)|x|−ap−1φ−1p−1|∂kφ∂tk|pp−1dxdt | (2.1) |
and
ω2(φ)=∫supp(φ)|x|−amp−mφ−mp−m|Lμφ|pp−mdxdt. | (2.2) |
We have the following a priori estimate.
Lemma 2.1. Let u∈Lploc(D) be a weak solution to (1.1) and (1.2). Then
−∫∂D1∂φ∂ν1f(x)dσdt≤C2∑i=1ωi(φ), | (2.3) |
for every φ∈V, provided that ωi(φ)<∞, i=1,2.
Proof. Let u∈Lploc(D) be a weak solution to (1.1) and (1.2) and φ∈V be such that ωi(φ)<∞, i=1,2. Then, by (1.11), there holds
∫D|x|a|u|pφdxdt−∫∂D1∂φ∂ν1f(x)dσdt≤∫D|u||∂kφ∂tk|dxdt+∫D|u|m|Lμφ|dxdt. | (2.4) |
Making use of Young's inequality, we obtain
∫D|u||∂kφ∂tk|dxdt=∫D(|x|ap|u|φ1p)(|x|−apφ−1p|∂kφ∂tk|)dxdt≤12∫D|x|a|u|pφdxdt+Cω1(φ). | (2.5) |
Similarly, we obtain
∫D|u|m|Lμφ|dxdt≤12∫D|x|a|u|pφdxdt+Cω2(φ). | (2.6) |
Therefore, combining (2.4)–(2.6), we obtain (2.3).
Let us introduce the function
F(x)=xN|x|αμ(1−|x|−(N+μ)−2αμ),x∈Ω, | (2.7) |
where the parameter αμ is given by (1.12). Elementary calculations show that
F≥0,LμF=0 in Ω,F|Γ0∪Γ1=0 | (2.8) |
and
∂F∂ν1|Γ1=−(N+μ+2αμ)xN<0,∂F∂ν0|Γ0=−|x|αμ(1−|x|−(N+μ)−2αμ)<0. | (2.9) |
Let ξ,ϑ,ι∈C∞(R) be three cut-off functions satisfying respectively
0≤ξ≤1,ξ(s)=1 if |s|≤1,ξ(s)=0 if |s|≥2, | (2.10) |
0≤ϑ≤1,ϑ(s)=1 if s≤0,ϑ(s)=0 if s≥1 | (2.11) |
and
ι≥0,supp(ι)⊂⊂(0,1). | (2.12) |
For sufficiently large T,R and ℓ, let
ιT(t)=ιℓ(tT),t>0, | (2.13) |
ξR(x)=F(x)ξℓ(|x|2R2),x∈Ω, | (2.14) |
ϑR(x)=F(x)ϑℓ(ln(|x|√R)ln(√R)),x∈Ω. | (2.15) |
Next, we consider functions of the form
φ(t,x)=ιT(t)ξR(x),(t,x)∈D | (2.16) |
and
ψ(t,x)=ιT(t)ϑR(x),(t,x)∈D. | (2.17) |
Lemma 2.2. For sufficiently large T,R and ℓ, the function φ defined by (2.16) belongs to V.
Proof. By (2.8), (2.10), (2.12)–(2.14) and (2.16), it can be easily seen that properties (i)–(iii) of Definition 1.1 are satisfied. Moreover, for (t,x)∈∂Di, i=0,1, one has
∂φ∂νi(t,x)=ιT(t)∂ξR∂νi(x)=ιT(t)∂F∂νi(x), | (2.18) |
which implies by (2.9) that
∂φ∂νi(t,x)≤0,(t,x)∈∂Di. |
This shows that property (iv) of Definition 1.1 is also satisfied. Therefore, φ∈V.
Similarly, using (2.8), (2.9), (2.11), (2.12), (2.15) and (2.17), we obtain the following result.
Lemma 2.3. For sufficiently large T,R and ℓ, the function ψ defined by (2.17) belongs to V.
For sufficiently large T,R and ℓ, let φ be the function defined by (2.16).
Lemma 2.4. The following estimate holds:
ω1(φ)≤CT1−kpp−1(lnR+Rαμ−ap−1+N+1). | (2.19) |
Proof. By (2.1) and (2.16), we obtain
ω1(φ)=(∫T0ι−1p−1T|dkιTdtk|pp−1dt)(∫1<|x|<√2R,xN>0|x|−ap−1ξR(x)dx). | (2.20) |
On the other hand, by (2.12) and (2.13), we obtain
|dkιTdtk|≤CT−kιℓ−k(tT),0<t<T, |
which yields
∫T0ι−1p−1T|dkιTdtk|pp−1dt≤CT−kpp−1∫T0ιℓ−kpp−1(tT)dt=CT1−kpp−1∫10ιℓ−kpp−1(s)ds, |
that is,
∫T0ι−1p−1T|dkιTdtk|pp−1dt≤CT1−kpp−1. | (2.21) |
Moreover, by (2.14), we have
∫1<|x|<√2R,xN>0|x|−ap−1ξR(x)dx=∫1<|x|<√2R,xN>0|x|−ap−1F(x)ξℓ(|x|2R2)dx. | (2.22) |
Using (2.7) and (2.10), we obtain
∫1<|x|<√2R,xN>0|x|−ap−1F(x)ξℓ(|x|2R2)dx≤∫1<|x|<√2R,xN>0|x|−ap−1F(x)dx≤∫1<|x|<√2R|x|αμ+1−ap−1dx=C∫√2Rr=1rαμ−ap−1+Ndr≤C(lnR+Rαμ−ap−1+N+1). | (2.23) |
Hence, in view of (2.20)–(2.23), we obtain (2.19).
Lemma 2.5. The following estimate holds:
ω2(φ)≤CTR(αμ+N−1)p−m(αμ+1+a+N)p−m. | (2.24) |
Proof. By (2.2) and (2.16), we have
ω2(φ)=(∫T0ιTdt)(∫1<|x|<√2R,xN>0|x|−amp−mξ−mp−mR|LμξR|pp−mdx). | (2.25) |
By (2.13), we obtain
∫T0ιTdt=∫T0ιℓ(tT)dt=T∫10ιℓ(s)ds, |
that is,
∫T0ιTdt=CT. | (2.26) |
Moreover, by (2.14), for <|x|<√2R,xN>0, we have
LμξR(x)=Lμ(F(x)ξℓ(|x|2R2))=Δ(F(x)ξℓ(|x|2R2))+μdiv((F(x)ξℓ(|x|2R2)x|x|2)=ξℓ(|x|2R2)ΔF(x)+F(x)Δ(ξℓ(|x|2R2))+2∇F(x)⋅∇(ξℓ(|x|2R2))+μξℓ(|x|2R2)div(F(x)x|x|2)+F(x)|x|2x⋅∇(ξℓ(|x|2R2))=ξℓ(|x|2R2)LμF(x)+F(x)Δ(ξℓ(|x|2R2))+(2∇F(x)+F(x)|x|2x)⋅∇(ξℓ(|x|2R2))=ξℓ(|x|2R2)LμF(x)+F(x)Δ(ξℓ(|x|2R2))+2ℓR−2|x|ξℓ−1(|x|2R2)ξ′(|x|2R2)(2∇F(x)⋅x|x|+|x|−1F(x)). |
In view of (2.8) (LμF=0), we obtain
LμξR(x)=F(x)Δ(ξℓ(|x|2R2))+2ℓR−2|x|ξℓ−1(|x|2R2)ξ′(|x|2R2)(2∇F(x)⋅x|x|+|x|−1F(x)), | (2.27) |
which implies by (2.10) that
∫1<|x|<√2R,xN>0|x|−amp−mξ−mp−mR|LμξR|pp−mdx=∫R<|x|<√2R,xN>0|x|−amp−mξ−mp−mR|LμξR|pp−mdx. | (2.28) |
On the other hand, by (2.7) and (2.10), for R<|x|<√2R,xN>0, we obtain
C1xNRαμ≤F(x)≤C2xNRαμ,|2∇F(x)⋅x|x|+|x|−1F(x)|≤CxNRαμ−1 | (2.29) |
and
|Δ(ξℓ(|x|2R2))|≤CR−2ξℓ−2(|x|2R2). | (2.30) |
Hence, in view of (2.27), (2.29), (2.30) and using that 0≤ξ≤1, there holds
|LμξR(x)|≤CxNRαμ−2ξℓ−2(|x|2R2),R<|x|<√2R,xN>0. | (2.31) |
Thus, using (2.14), (2.28), (2.29) and (2.31), we get
∫1<|x|<√2R,xN>0|x|−amp−mξ−mp−mR|LμξR|pp−mdx≤CR(αμ−2)pp−m∫R<|x|<√2R,xN>0|x|−amp−mF−mp−m(x)xpp−mNξℓ−2pp−m(|x|2R2)dx≤CR(αμ−2)p−αμmp−m∫R<|x|<√2R,xN>0xN|x|−amp−mdx≤CR(αμ−2)p−αμmp−m∫R<|x|<√2R|x|1−amp−mdx≤CR(αμ−2)p−αμmp−mR1−amp−mRN, |
that is,
∫1<|x|<√2R,xN>0|x|−amp−mξ−mp−mR|LμξR|pp−mdx≤CR(αμ+N−1)p−m(αμ+1+a+N)p−m. | (2.32) |
Finally, (2.24) follows from (2.25), (2.26) and (2.32).
For sufficiently large T,R and ℓ, let ψ be the function defined by (2.17).
Lemma 2.6. The following estimate holds:
ω1(ψ)≤CT1−kpp−1(lnR+Rαμ−ap−1+N+1). | (2.33) |
Proof. By (2.1) and (2.17), we obtain
ω1(ψ)=(∫T0ι−1p−1T|dkιTdtk|pp−1dt)(∫1<|x|<R,xN>0|x|−ap−1ϑR(x)dx). | (2.34) |
Moreover, by (2.15), we have
∫1<|x|<R,xN>0|x|−ap−1ϑR(x)dx=∫1<|x|<R,xN>0|x|−ap−1F(x)ϑℓ(ln(|x|√R)ln(√R))dx. | (2.35) |
Using (2.7) and (2.11), we obtain
∫1<|x|<R,xN>0|x|−ap−1F(x)ϑℓ(ln(|x|√R)ln(√R))dx≤∫1<|x|<R,xN>0|x|−ap−1F(x)dx≤C(lnR+Rαμ−ap−1+N+1). | (2.36) |
Hence, in view of (2.21), (2.34)–(2.36), we obtain (2.33).
Lemma 2.7. Let (αμ+N−1)p=m(αμ+1+a+N). Then, the following estimate holds:
ω2(ψ)≤CT(lnR)−mp−m. | (2.37) |
Proof. By (2.2) and (2.17), we have
ω2(ψ)=(∫T0ιTdt)(∫1<|x|<R,xN>0|x|−amp−mϑ−mp−mR|LμϑR|pp−mdx). | (2.38) |
Similar calculations to those done in the proof of Lemma 2.5 give us
LμϑR(x)=F(x)Δ(ϑℓ(ln(|x|√R)ln(√R)))+ℓln(√R)|x|ϑℓ−1(ln(|x|√R)ln(√R))ϑ′(ln(|x|√R)ln(√R))(2∇F(x)⋅x|x|+|x|−1F(x)), | (2.39) |
which implies by (2.11) that
∫1<|x|<R,xN>0|x|−amp−mϑ−mp−mR|LμϑR|pp−mdx=∫√R<|x|<R,xN>0|x|−amp−mϑ−mp−mR|LμϑR|pp−mdx. | (2.40) |
Moreover, by (2.7) and (2.11), we obtain, as |x|→∞,
C1xN|x|αμ≤F(x)≤C2xN|x|αμ,|2∇F(x)⋅x|x|+|x|−1F(x)|≤CxN|x|αμ−1 | (2.41) |
and
|Δ(ϑℓ(ln(|x|√R)ln(√R)))|≤C(lnR)−1|x|−2ϑℓ−2(ln(|x|√R)ln(√R)),√R<|x|<R,xN>0. | (2.42) |
In view of (2.39), (2.41), (2.42) and using that 0≤ϑ≤1, we get
|LμϑR(x)|≤CxN|x|αμ−2(lnR)−1ϑℓ−2(ln(|x|√R)ln(√R)),√R<|x|<R,xN>0. | (2.43) |
Next, it follows from (2.40), (2.41) and (2.43) that
∫1<|x|<R,xN>0|x|−amp−mϑ−mp−mR|LμϑR|pp−mdx≤C(lnR)−pp−m∫√R<|x|<R,xN>0|x|(αμ−2)p−m(a+αμ)p−mxNϑℓ−2pp−m(ln(|x|√R)ln(√R))dx≤C(lnR)−pp−m∫√R<|x|<R|x|(αμ−1)p−m(a+αμ+1)p−mdx. |
Using that (αμ+N−1)p=m(α+1+a+N), we get
∫1<|x|<R,xN>0|x|−amp−mϑ−mp−mR|LμϑR|pp−mdx≤C(lnR)−pp−m∫√R<|x|<R|x|−Ndx=C(lnR)−pp−m∫Rr=√Rr−1dr≤C(lnR)−mp−m. | (2.44) |
Finally, (2.37) follows from (2.26), (2.38) and (2.44).
We use the contradiction argument. Namely, we suppose that u∈Lploc(D) is a weak solutions to (1.1) and (1.2). We first consider the case
p>m,(αμ+N−1)p−m(αμ+1+a+N)<0. | (3.1) |
By Lemmas 2.1 and 2.2, for sufficiently large T,R and ℓ, there holds
−∫∂D1∂φ∂ν1f(x)dσdt≤C2∑i=1ωi(φ), | (3.2) |
where φ is the function defined by (2.16). On the other hand, by (2.9), (2.18) and (2.26), we have
−∫∂D1∂φ∂ν1f(x)dσdt=−∫∂D1ιT(t)f(x)∂F∂ν1(x)dσdt=(N+μ+2αμ)(∫T0ιT(t)dt)∫Γ1f(x)xNdσ=CT∫Γ1f(x)xNdσ. | (3.3) |
Then, using Lemmas 2.4 and 2.5, (3.2) and (3.3), we obtain
T∫Γ1f(x)xNdσ≤C(T1−kpp−1(lnR+Rαμ−ap−1+N+1)+TR(αμ+N−1)p−m(αμ+1+a+N)p−m), |
that is,
∫Γ1f(x)xNdσ≤C(T−kpp−1(lnR+Rαμ−ap−1+N+1)+R(αμ+N−1)p−m(αμ+1+a+N)p−m). |
Next, taking T=Rθ, where
θ>max{0,p−1kp(αμ−ap−1+N+1)}, | (3.4) |
the above estimate reduces to
∫Γ1f(x)xNdσ≤C(R−θkpp−1lnR+Rζ1+Rζ2), | (3.5) |
where
ζ1=αμ−ap−1+N+1−θkpp−1,ζ2=(αμ+N−1)p−m(αμ+1+a+N)p−m. |
Notice that due to (3.4), one has ζ1<0. Moreover, by (3.1), we get ζ2<0. Therefore, passing to the limit as R→∞ in (3.5), we obtain ∫Γ1f(x)xNdσ≤0, which contradicts (1.13).
Next, we consider the case
p>m,(αμ+N−1)p−m(αμ+1+a+N)=0. | (3.6) |
By Lemmas 2.1 and 2.3, for sufficiently large T,R and ℓ, there holds
−∫∂D1∂ψ∂ν1f(x)dσdt≤C2∑i=1ωi(ψ), | (3.7) |
where ψ is the function defined by (2.17). As in the previous case, using Lemmas 2.6 and 2.7, (2.9), (2.17) and (3.7), we obtain
T∫Γ1f(x)xNdσ≤C(T1−kpp−1(lnR+Rαμ−ap−1+N+1)+T(lnR)−mp−m), |
that is,
∫Γ1f(x)xNdσ≤C(T−kpp−1(lnR+Rαμ−ap−1+N+1)+(lnR)−mp−m). | (3.8) |
Hence, taking T=Rθ, where the parameter θ satisfies (3.4), and passing to the limit as R→∞ in (3.8), we reach a contradiction with (1.13). This completes the proof of part (Ⅰ) of Theorem 1.3.
Assume that (1.15) holds. Let us consider a parameter δ satisfying
max{−μ−αμ,1+m(a+2)p−m,1}<δ<N+αμ. | (3.9) |
Notice that −μ−αμ<N+αμ and 1<N+αμ. Moreover, due to (1.15), one has 1+m(a+2)p−m<N+αμ. Hence, the set of δ satisfying (3.9) is nonempty. Let
0<ε<[(N+αμ−δ)(δ+μ+αμ)]1p−m. | (3.10) |
We consider functions of the form
uδ,ε(x)=εx1mN|x|−δm,x∈Ω. | (3.11) |
Elementary calculations show that
−Δumδ,ε+μ|x|2x⋅∇umδ,ε=εm(N+αμ−δ)(δ+μ+αμ)xN|x|−δ−2,x∈Ω. |
Hence, using (3.9)–(3.11), for all x∈Ω, we obtain
−Δumδ,ε+μ|x|2x⋅∇umδ,ε=(|x|aεpxpmN|x|−δpm)εm−p(N+αμ−δ)(δ+μ+αμ)x1−pmN|x|−δ−2−a+δpm=|x|aupδ,ε(x)εm−p(N+αμ−δ)(δ+μ+αμ)x1−pmN|x|−δ−2−a+δpm≥|x|aupδ,ε(x)|x|(δ−1)(pm−1)−(a+2)≥|x|aupδ,ε(x). |
Therefore, uδ,ε is a stationary solution to (1.1) and (1.2) with f(x)=εmxN, x∈Γ1. This completes the proof of part (Ⅱ) of Theorem 1.3.
We investigated The existence and nonexistence of weak solutions to the evolution inequality (1.1) under the Dirichlet-type boundary conditions (1.2). When a≤−2, we proved that (1.1) and (1.2) admit no critical behavior, namely, for all p>m≥1, (1.1) and (1.2) admit stationary solutions for some f>0. When a>−2, we proved that (1.1) and (1.2) admit a critical exponent
p∗=p∗(m,a,μ,N)=m+m(a+2)αμ+N−1, |
in the following sense:
(ⅰ) If
m<p≤p∗, |
then (1.1) and (1.2) admit no weak solution, provided that f∈L1(Γ1) and
∫Γ1f(x)xNdσ>0. |
(ⅱ) If
p>p∗, |
then (1.1) and (1.2) admit (stationary) solutions, for some f>0.
It is interesting to observe that in the case a>−2, the critical exponent p∗ depends only on m,a,μ and N, but it is independent of k, the order of the time-derivative. Therefore, our obtained results hold in both parabolic and hyperbolic cases. Finally, let us mention that comparing with previous existing results in the literature, in this study no restriction on the sign of solutions is imposed.
The authors extend their appreciation to the Deanship of Scientific Research at the Imam Mohammad Ibn Saud Islamic University (IMSIU) for funding and supporting this work through Research Partnership Program no. RP-21-09-02.
The authors declare no conflict of interest.
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