Global existence and blow-up of solutions to pseudo-parabolic equation for Baouendi-Grushin operator

Aishabibi Dukenbayeva

doi:10.1515/math-2025-0144

Article Open Access

Global existence and blow-up of solutions to pseudo-parabolic equation for Baouendi-Grushin operator

Aishabibi Dukenbayeva

Published/Copyright: May 23, 2025

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

$Open Mathematics$

From the journal Open Mathematics Volume 23 Issue 1

Abstract

In this note, we study a global existence and blow-up of the positive solutions to the initial-boundary value problem of the nonlinear pseudo-parabolic equation for the Baouendi-Grushin operator. The approach is based on the concavity argument and the Poincaré inequality related to the Baouendi-Grushin operator from [Suragan and Yessirkegenov, Sharp remainder of the Poincaré inequality for Baouendi–Grushin vector fields, Asian-Eur. J. Math. 16 (2023), 2350041], inspired by the recent work [Ruzhansky et al., Global existence and blow-up of solutions to porous medium equation and pseudo-parabolic equation, I. Stratified group, Manuscripta Math. 171 (2023), 377–395].

Keywords: blow-up; global solution; pseudo-parabolic equation; Baouendi-Grushin operator

MSC 2010: 35K65; 35K91; 35B44; 35A01

1 Introduction

Consider the initial-boundary problem of the nonlinear pseudo-parabolic equation

(1) u t − Δ u t − Δ u = f ( u ) , ( x , y ) ∈ Ω , t > 0 , u ( x , y , t ) = 0 , ( x , y ) ∈ ∂ Ω , t > 0 , u ( x , y , 0 ) = u 0 ( x , y ) ≥ 0 , ( x , y ) ∈ Ω ¯ ,

where f is locally Lipschitz continuous on R , f ( 0 ) = 0 , and such that f ( u ) > 0 for u > 0 , where Ω is a smoothly bounded domain in R n . Here, u 0 is a non-negative and non-trivial function in C 1 ( Ω ¯ ) with u 0 ( x , y ) = 0 on the boundary ∂ Ω .

The energy of isotropic materials can be described using a pseudo-parabolic equation [1]. Pseudo-parabolic equations are also used to model certain wave processes [2], as well as the filtration of two-phase flow in porous media considering dynamic capillary pressure [3]. We also refer to the works [4–6] by Ting, Showalter, and Gopala Rao on the homogeneous problem (1) with f = 0 , after which considerable attention has been paid to the study of nonlinear pseudo-parabolic equations. As for the inhomogeneous nonlinear problem (1) with f ( u ) = u p , we can refer to Benedetto and Pierre [7] for the maximum principle and to Cao et al. [8] for the critical global existence exponent and for the critical Fujita exponent. Nowadays, numerous researchers have investigated the global existence and finite-time blow-up of solutions to pseudo-parabolic equations in both bounded and unbounded domains, as referenced in works such as [9–16] and others.

In this note, we consider an extension of problem (1) to the Baouendi-Grushin setting.

Let z ≔ ( x , y ) ≔ ( x 1 , … , x m , y 1 , … , y k ) ∈ R m × R k with m , k ≥ 1 and m + k = n . In this setting, we define the corresponding sub-elliptic gradient on R m + k by

(2) ∇ γ ≔ ( X 1 , … , X m , Y 1 , … , Y k ) = ( ∇ x , ∣ x ∣ γ ∇ y )

and Baouendi-Grushin operator by

(3) Δ γ = ∑ i = 1 m X i 2 + ∑ j = 1 k Y j 2 = Δ x + ∣ x ∣ 2 γ Δ y = ∇ γ ⋅ ∇ γ ,

where

X i = ∂ ∂ x i , i = 1 , … , m , Y j = ∣ x ∣ γ ∂ ∂ y j , γ ≥ 0 , j = 1 , … , k .

Now, we are ready to state our problem in the Baouendi-Grushin setting: Let D ⊂ R m + k be a bounded domain (open and connected) supporting the divergence formula and D \ { ( x , y ) ∈ D ¯ : x = 0 } consists of only one connected component. We consider the problem

(4) u t − Δ γ u t − Δ γ u = f ( u ) , ( x , y ) ∈ D , t > 0 , u ( x , y , t ) = 0 , ( x , y ) ∈ ∂ D , t > 0 , u ( x , y , 0 ) = u 0 ( x , y ) ≥ 0 , ( x , y ) ∈ D ¯ ,

where f is locally Lipschitz continuous on R , f ( 0 ) = 0 , and such that f ( u ) > 0 for u > 0 . As mentioned earlier, we assume that u 0 is a non-negative and non-trivial function in C 1 ( D ¯ ) with u 0 ( x , y ) = 0 on the boundary ∂ D .

Note that our problem (4) is a degenerate extension of problem (1). Namely, when γ = 0 , the pseudo-parabolic problem (4) reduces to problem (1). In this note, we are interested in global existence and blow-up of the positive solutions to the degenerate problem (4). The known methods for problem (1) cannot be directly applied to problem (4) because of its degeneracy. Here, we show how the degeneracy parameter γ influences to the global existence and blow-up property of the positive solutions of problem (4).

Veron and Pohozaev in [17] studied the blow-up phenomena for the following equation on the Heisenberg group H n :

(5) ∂ u ( x , t ) ∂ t − ℒ u ( x , t ) = ∣ u ( x , t ) ∣ p , ( x , t ) ∈ H n × ( 0 , + ∞ ) ,

where ℒ is the sub-Laplacian on H n . We can refer to [18–22] for blow-up type results for semilinear diffusion and pseudo-parabolic equations on the Heisenberg group as well as to [23] for the Fujita exponent on general unimodular Lie groups. We also refer to Brill [24] and David and Jet [25] for singular pseudo-parabolic equations and degenerate pseudo-parabolic equations, where the authors obtained existence and uniqueness results. Recently, in [26], we studied global existence and blow-up type properties for problem (5) with the Baouendi-Grushin operator Δ γ instead of the sub-Laplacian ℒ . Here, our research is inspired by the work [27] where the authors obtained global existence and blow-up type results for problem (4) with a sub-Laplacian instead of the Baouendi-Grushin operator on stratified Lie groups.

However, unlike the sub-Laplacian, the Baouendi-Grushin operator that we will deal in this note is a sum of squares of smooth vector fields satisfying the following Hörmander rank condition only for an even positive integer γ :

rank Lie [ X 1 , … , X m , Y 1 , … , Y k ] = n .

So, from (3), one can note that unlike the sub-Laplacian on the Heisenberg group or unimodular Lie groups, the Baouendi-Grushin operator not always can be represented by sum of squares of Hörmander’s vector fields.

Recall also that the anisotropic dilation attached to the Baouendi-Grushin operator is defined by

δ λ ( z ) = ( λ x , λ 1 + γ y ) , λ > 0 ,

and the homogeneous dimension with respect to δ λ is defined by

(6) Q = m + ( 1 + γ ) k .

A change of variables formula for the Lebesgue measure implies that

d ∘ δ λ ( x , y ) = λ Q d x d y .

Let H 0 1 , γ ( D ) be the Sobolev space obtained as completion of C 0 ∞ ( D ) with respect to the norm

‖ f ‖ H 0 1 , γ ( D ) ≔ ∫ D ∣ ∇ γ f ∣ 2 d x d y 1 2 .

Thus, the first result of this note on the blow-up property takes the form:

Theorem 1

Assume that

(7) α F ( u ) ≤ u f ( u ) + β u 2 + α θ , u > 0 ,

where

F ( u ) = ∫ 0 u f ( s ) d s ,

for some

(8) α > 2 and 0 < β ≤ λ 1 ( α − 2 ) 2 θ > 0 , ,

and λ 1 isndi-Grushin operator on D. Assume also that the initial data u 0 ∈ L ∞ ( D ) ∩ H 0 1 , γ ( D ) satisfies

(9) ℱ 0 ≔ − 1 2 ∫ D ∣ ∇ γ u 0 ( x , y ) ∣ 2 d x d y + ∫ D ( F ( u 0 ( x , y ) ) − θ ) d x d y > 0 .

Then any positive solution u of (4) blows up in finite time T * , that is, there exists

(10) 0 < T * ≤ M σ ∫ D ( u 0 2 + ∣ ∇ γ u 0 ∣ 2 ) d x d y ,

such that

(11) lim t → T * ∫ 0 t ∫ D [ u 2 + ∣ ∇ γ u ∣ 2 ] d x d y d τ = + ∞ ,

where σ = α 2 − 1 > 0 and

M = ( 1 + σ ) 1 + 1 σ ∫ D ( u 0 2 + ∣ ∇ γ u 0 ∣ 2 ) d x d y 2 2 α ℱ 0 .

Remark 1

The authors in [28] used the same condition on f ( u ) for a parabolic equation with the classical Laplacian (see also [29,30] for particular cases). We can also refer to [27] and [31,32] for recent papers on such conditions.

Remark 2

For more details on spectral properties of the Dirichlet Baouendi-Grushin operator, we refer to, e.g., [33–35].

The following result indicates that, for certain nonlinearities, positive solutions can be controlled when they exist.

Theorem 2

Assume that

(12) α F ( u ) ≥ u f ( u ) + β u 2 + α θ , u > 0 ,

where

F ( u ) = ∫ 0 u f ( s ) d s ,

for some

(13) β ≥ 2 − α 2 and α ≤ 0 , θ ≥ 0 .

Let the initial data u 0 ∈ L ∞ ( D ) ∩ H 0 1 , γ ( D ) satisfy

(14) ℱ 0 ≔ − 1 2 ∫ D ∣ ∇ γ u 0 ( x , y ) ∣ 2 d x d y + ∫ D ( F ( u 0 ( x , y ) ) − θ ) d x d y > 0 .

If u is a positive local solution of problem (4), then it is global with the property

∫ D ( u 2 + ∣ ∇ γ u ∣ 2 ) d x d y ≤ exp ( − ( 2 − α ) t ) ∫ D ( u 0 2 + ∣ ∇ γ u 0 ∣ 2 ) d x d y .

The following Poincaré inequality established in [36] plays an important role for our analysis:

Lemma 1

Let D ⊂ R m + k be a bounded domain (open and connected) supporting the divergence formula and D \ { ( x , y ) ∈ D ¯ : x = 0 } consists of only one connected component. For every function u ∈ H 0 1 , γ ( D ) , we have

(15) ∫ D ∣ ∇ γ u ∣ 2 d x d y ≥ λ 1 ∫ D ∣ u ∣ 2 d x d y ,

where λ 1 is the first eigenvalue of the Dirichlet Baouendi-Grushin operator on D.

2 Proofs

Let us begin with the proof of Theorem 1 on the blow-up property of problem (4).

Denote

(16) E ( t ) ≔ ∫ 0 t ∫ D ( u 2 + ∣ ∇ γ u ∣ 2 ) d x d y d τ + M , t ≥ 0 ,

where M is a positive constant to be chosen later. A direct calculation implies that

E ′ ( t ) = ∫ D ( u 2 + ∣ ∇ γ u ∣ 2 ) d x d y = ∫ 0 t d d τ ∫ D ( u 2 + ∣ ∇ γ u ∣ 2 ) d x d y d τ + ∫ D ( u 0 2 + ∣ ∇ γ u 0 ∣ 2 ) d x d y .

It implies for arbitrary δ > 0 that

(17) ( E ′ ( t ) ) 2 ≤ ( 1 + δ ) ∫ 0 t d d τ ∫ D ( u 2 + ∣ ∇ γ u ∣ 2 ) d x d y d τ 2 + 1 + 1 δ ∫ D ( u 0 2 + ∣ ∇ γ u 0 ∣ 2 ) d x d y 2 .

By making use of (7) and integration by parts, we obtain for E ″ ( t ) that

E ″ ( t ) = 2 ∫ D u u t d x d y + ∫ D ( ∣ ∇ γ u ∣ 2 ) t d x d y = 2 ∫ D ( u Δ γ u + u ∇ γ ⋅ ( ∇ γ u t ) + u f ( u ) ) d x d y + ∫ D ( ∣ ∇ γ u ∣ 2 ) t d x d y = − 2 ∫ D ( ∣ ∇ γ u ∣ 2 + ∇ γ u ⋅ ∇ γ u t ) d x d y + 2 ∫ D u f ( u ) d x d y + ∫ D ( ∣ ∇ γ u ∣ 2 ) t d x d y ≥ − 2 ∫ D ∣ ∇ γ u ∣ 2 d x d y + 2 ∫ D ( α F ( u ) − β u 2 − α θ ) d x d y = 2 α − 1 2 ∫ D ∣ ∇ γ u ∣ 2 d x d y + ∫ D ( F ( u ) − θ ) d x d y + 2 ( α − 2 ) 2 ∫ D ∣ ∇ γ u ∣ 2 d x d y − 2 β ∫ D u 2 d x d y ≥ 2 α − 1 2 ∫ D ∣ ∇ γ u ∣ 2 d x d y + ∫ D ( F ( u ) − θ ) d x d y + 2 λ 1 ( α − 2 ) 2 − β ∫ D u 2 d x d y ≥ 2 α − 1 2 ∫ D ∣ ∇ γ u ∣ 2 d x d y + ∫ D ( F ( u ) − θ ) d x d y ≕ 2 α ℱ ( t ) ,

where we have applied assumption (8) and the Poincaré inequality from Lemma 1 in the last two estimates above. Note that for ℱ 0 from (9), we have ℱ ( 0 ) = ℱ 0 . Then, taking into account

ℱ ( t ) = ℱ ( 0 ) + ∫ 0 t d ℱ ( τ ) d τ d τ ,

we continue the aforementioned estimation as follows:

(18) E ″ ( t ) ≥ 2 α ℱ ( t ) = 2 α ℱ ( 0 ) + ∫ 0 t d ℱ ( τ ) d τ d τ = 2 α ℱ ( 0 ) − 1 2 ∫ 0 t ∫ D d d τ ∣ ∇ γ u ∣ 2 d x d y d τ + ∫ 0 t ∫ D d d τ ( F ( u ) − θ ) d x d y d τ = 2 α ℱ ( 0 ) − ∫ 0 t ∫ D ∣ ∇ u ⋅ ∇ γ u τ d x d y d τ + ∫ 0 t ∫ D F u ( u ) u τ d x d y d τ = 2 α ℱ ( 0 ) + ∫ 0 t ∫ D [ Δ γ u + f ( u ) ] u τ d x d y d τ = 2 α ℱ ( 0 ) + ∫ 0 t ∫ D u τ 2 − u τ ∇ γ ⋅ ( ∇ γ u τ ) d x d y d τ = 2 α ℱ ( 0 ) + ∫ 0 t ∫ D u τ 2 + ∣ ∇ γ u τ ∣ 2 d x d y d τ .

By combining (16), (17), (18), and taking σ = δ = α 2 − 1 > 0 , we obtain

(19) E ″ ( t ) E ( t ) − ( 1 + σ ) ( E ′ ( t ) ) 2 ≥ 2 α M ℱ ( 0 ) + 2 α ∫ 0 t ∫ D ( u τ 2 + ∣ ∇ γ u τ ∣ 2 ) d x d y d τ ∫ 0 t ∫ D ( u 2 + ∣ ∇ γ u ∣ 2 ) d x d y d τ − ( 1 + σ ) ( 1 + δ ) ∫ 0 t d d τ ∫ D ( u 2 + ∣ ∇ γ u ∣ 2 ) d x d y d τ 2 − ( 1 + σ ) 1 + 1 δ ∫ D ( u 0 2 + ∣ ∇ γ u 0 ∣ 2 ) d x d y 2 = 2 α M ℱ ( 0 ) − ( 1 + σ ) 1 + 1 δ ∫ D ( u 0 2 + ∣ ∇ γ u 0 ∣ 2 ) d x d y 2 + 2 α ∫ 0 t ∫ D ( u τ 2 + ∣ ∇ γ u τ ∣ 2 ) d x d y d τ ∫ 0 t ∫ D ( u 2 + ∣ ∇ γ u ∣ 2 ) d x d y d τ − ∫ 0 t ∫ D ( u u τ + ∇ γ u ⋅ ∇ γ u τ ) d x d y d τ 2 .

Here, let us first show that

(20) ∫ 0 t ∫ D ( u 2 + ∣ ∇ γ u ∣ 2 ) d x d y d τ ∫ 0 t ∫ D ( u τ 2 + ∣ ∇ γ u τ ∣ 2 ) d x d y d τ − ∫ 0 t ∫ D ( u u τ + ∇ γ u ⋅ ∇ γ u τ ) d x d y d τ 2 ≥ 0 .

For this, by using Hölder’s and Cauchy-Schwarz inequalities, we observe that

(21) ∫ 0 t ∫ D ( u u τ + ∇ γ u ⋅ ∇ γ u τ ) d x d y d τ 2 ≤ ∫ D ∫ 0 t u 2 d τ 1 2 ∫ 0 t u τ 2 d τ 1 2 d x d y + ∫ D ∫ 0 t ∣ ∇ γ u ∣ 2 d τ 1 2 ∫ 0 t ∣ ∇ γ u τ ∣ 2 d τ 1 2 d x d y 2 = ∫ D ∫ 0 t u 2 d τ 1 2 ∫ 0 t u τ 2 d τ 1 2 d x d y 2 + ∫ D ∫ 0 t ∣ ∇ γ u ∣ 2 d τ 1 2 ∫ 0 t ∣ ∇ γ u τ ∣ 2 d τ 1 2 d x d y 2 + 2 ∫ D ∫ 0 t u 2 d τ 1 2 ∫ 0 t u τ 2 d τ 1 2 d x d y ∫ D ∫ 0 t ∣ ∇ γ u ∣ 2 d τ 1 2 ∫ 0 t ∣ ∇ γ u τ ∣ 2 d τ 1 2 d x d y ≤ ∫ D ∫ 0 t u 2 d τ d x d y ∫ D ∫ 0 t u τ 2 d τ d x d y + ∫ D ∫ 0 t ∣ ∇ γ u ∣ 2 d τ d x d y ∫ D ∫ 0 t ∣ ∇ γ u τ ∣ 2 d τ d x d y + 2 ∫ D ∫ 0 t u 2 d τ d x d y ∫ D ∫ 0 t u τ 2 d τ d x d y ∫ D ∫ 0 t ∣ ∇ γ u ∣ 2 d τ d x d y ∫ D ∫ 0 t ∣ ∇ γ u τ ∣ 2 d τ d x d y 1 2 .

Then, by taking into account (21), we can verify (20) as follows:

∫ 0 t ∫ D ( u 2 + ∣ ∇ γ u ∣ 2 ) d x d y d τ ∫ 0 t ∫ D ( u τ 2 + ∣ ∇ γ u τ ∣ 2 ) d x d y d τ − ∫ 0 t ∫ D ( u u τ + ∇ γ u ⋅ ∇ γ u τ ) d x d y d τ 2 ≥ ∫ D ∫ 0 t u 2 d τ d x d y 1 2 ∫ D ∫ 0 t ∣ ∇ γ u τ ∣ 2 d τ d x d y 1 2 − ∫ D ∫ 0 t ∣ ∇ γ u ∣ 2 d τ d x d y 1 2 ∫ D ∫ 0 t u τ 2 d τ d x d y 1 2 2 ≥ 0 .

Thus, by using (20) in (19), we arrive at

E ″ ( t ) E ( t ) − ( 1 + σ ) ( E ′ ( t ) ) 2 ≥ 2 α M ℱ ( 0 ) − ( 1 + σ ) 1 + 1 δ ∫ D ( u 0 2 + ∣ ∇ γ u 0 ∣ 2 ) d x d y 2 ,

which taking into account ℱ ( 0 ) > 0 implies

(22) E ″ ( t ) E ( t ) − ( 1 + σ ) ( E ′ ( t ) ) 2 ≥ 0 ,

if we choose M as follows:

M = ( 1 + σ ) 1 + 1 δ ∫ D ( u 0 2 + ∣ ∇ γ u 0 ∣ 2 ) d x d y 2 2 α ℱ ( 0 ) .

From (22), one can derive for t ≥ 0 that

d d t E ′ ( t ) E σ + 1 ( t ) ≥ 0 ⇒ E ′ ( t ) ≥ E ′ ( 0 ) E σ + 1 ( 0 ) E 1 + σ ( t ) , E ( 0 ) = M .

Here, for σ = α 2 − 1 > 0 , we conclude that

E ( t ) ≥ 1 M σ − σ ∫ D ( u 0 2 + ∣ ∇ γ u 0 ∣ 2 ) d x d y M σ + 1 t − 1 σ ,

hence the blow-up time T * satisfies

0 < T * ≤ M σ ∫ D ( u 0 2 + ∣ ∇ γ u 0 ∣ 2 ) d x d y ,

as desired.

Let us now prove the global existence result.

Proof

Denote

ℰ ( t ) ≔ ∫ D ( u 2 + ∣ ∇ γ u ∣ 2 ) d x d y .

Recalling the functional ℱ ( t ) from the proof of Theorem 1 and using (12), we have

ℰ ′ ( t ) = 2 ∫ D u u t d x d y + ∫ D ( ∣ ∇ γ u ∣ 2 ) t d x d y = 2 ∫ D ( u Δ γ u + u ∇ γ ⋅ ( ∇ γ u t ) + u f ( u ) ) d x d y + ∫ D ( ∣ ∇ γ u ∣ 2 ) t d x d y = − 2 ∫ D ( ∣ ∇ γ u ∣ 2 + ∇ γ u ⋅ ∇ γ u t ) d x d y + 2 ∫ D u f ( u ) d x d y + ∫ D ( ∣ ∇ γ u ∣ 2 ) t d x d y ≤ 2 α − 1 2 ∫ D ∣ ∇ γ u ∣ 2 d x d y + ∫ D ( F ( u ) − θ ) d x d y − 2 ( 2 − α ) 2 ∫ D ∣ ∇ γ u ∣ 2 d x d y − 2 β ∫ D u 2 d x d y ≤ 2 α − 1 2 ∫ D ∣ ∇ γ u ∣ 2 d x d y + ∫ D ( F ( u ) − θ ) d x d y − ( 2 − α ) ( ℰ ( t ) − ∫ D u 2 d x d y ) − 2 β ∫ D u 2 d x d y , = 2 α ℱ ( t ) − ( 2 − α ) ℰ ( t ) + ( 2 − α − 2 β ) ∫ D u 2 d x d y .

By using the calculation from (18) and β ≥ 2 − α 2 , we conclude that

ℰ ′ ( t ) + ( 2 − α ) ℰ ( t ) ≤ 2 α ℱ 0 + ∫ 0 t ∫ D ( u τ 2 + ∣ ∇ γ u τ ∣ 2 ) d x d y d τ ≤ 0 ,

which means

ℰ ( t ) ≤ exp ( − ( 2 − α ) t ) ℰ ( 0 ) ,

as desired.□

Funding information: This research was funded by the Committee of Science of the Ministry of Science and Higher Education of the Republic of Kazakhstan (Grant No. AP14972714).
Author contributions: The author confirms the sole responsibility for the conception of the study, presented results, and preparation of the manuscript.
Conflict of interest: The author states no conflict of interest.

References

[1] P. J. Chen and M. E. Gurtin, On a theory of heat conduction involving two temperatures, Z. Angew. Math. Phys. 19 (1968), 614–627. 10.1007/BF01594969Search in Google Scholar

[2] T. B. Benjamin, J. L. Bona, and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Roy. Soc. A 272 (1972), 47–78. 10.1098/rsta.1972.0032Search in Google Scholar

[3] G. I. Barenblatt, J. Garcia-Azorero, A. De Pablo, and J. L. Vazquez, Mathematical model of the non-equilibrium water-oil displacement in porous strata, Appl. Anal. 65 (1997), 19–45. 10.1080/00036819708840547Search in Google Scholar

[4] T. W. Ting, Parabolic and pseudo-parabolic partial differential equations, J. Math. Soc. Japan 21 (1969), 440–453. 10.2969/jmsj/02130440Search in Google Scholar

[5] R. E. Showalter and T. W. Ting, Pseudoparabolic partial differential equations, SIAM J. Math. Anal. 1 (1970), 1–26. 10.1137/0501001Search in Google Scholar

[6] V. R. Gopala Rao and T. W. Ting, Solutions of pseudo-heat equations in the whole space, Arch. Ration. Mech. Anal. 49 (1972), 57–78. 10.1007/BF00281474Search in Google Scholar

[7] E. D. Benedetto and M. Pierre, On the maximum principle for pseudoparabolic equations, Indiana Univ. Math. J. 30 (1981), 821–854. 10.1512/iumj.1981.30.30062Search in Google Scholar

[8] Y. Cao, J. X. Yin, and C. P. Wang, Cauchy problems of semilinear pseudo-parabolic equations, J. Differential Equations 246 (2009), 4568–4590. 10.1016/j.jde.2009.03.021Search in Google Scholar

[9] M. O. Korpusov and A. G. Sveshnikov, On the blow-up of solutions of semilinear pseudoparabolic equations with rapidly growing nonlinearities, Zh. Vychisl. Mat. Mat. Fiz. 45 (2005), 145–155 (in Russian). Search in Google Scholar

[10] M. O. Korpusov and A. G. Sveshnikov, On the blow-up in a finite time of solutions of initial-boundary-value problems for pseudoparabolic equations with the pseudo-Laplacian, Zh. Vychisl. Mat. Mat. Fiz. 45 (2005), 272–286 (in Russian). Search in Google Scholar

[11] Q. F. Long and J. Q. Chen, Blow-up phenomena for a nonlinear pseudo-parabolic equation with nonlocal source, Appl. Math. Lett. 74 (2017), 181–186. 10.1016/j.aml.2017.06.006Search in Google Scholar

[12] P. Luo, Blow-up phenomena for a pseudo-parabolic equation, Math. Methods Appl. Sci. 38 (2015), 2636–2641. 10.1002/mma.3253Search in Google Scholar

[13] X. M. Peng, Y. D. Shang, and X. X. Zheng, Blow-up phenomena for some nonlinear pseudo-parabolic equations, Appl. Math. Lett. 56 (2016), 17–22. 10.1016/j.aml.2015.12.005Search in Google Scholar

[14] R. Z. Xu and J. Su, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal. 264 (2013), 2732–2763. 10.1016/j.jfa.2013.03.010Search in Google Scholar

[15] R. Z. Xu, X. C. Wang, and Y. B. Yang, Blowup and blowup time for a class of semilinear pseudo-parabolic equations with high initial energy, Appl. Math. Lett. 83 (2018), 176–181. 10.1016/j.aml.2018.03.033Search in Google Scholar

[16] R. Z. Xu and X. C. Wang, Global existence and finite time blowup for a nonlocal semilinear pseudo-parabolic equation, Adv. Nonlinear Anal. 10 (2021), 261–288. 10.1515/anona-2020-0141Search in Google Scholar

[17] L. Véron and S. I. Pohozaev, Nonexistence results of solutions of semilinear differential inequalities on the Heisenberg group, Manuscripta Math. 102 (2000), 85–99. 10.1007/PL00005851Search in Google Scholar

[18] B. Ahmad, A. Alsaedi, M. Kirane, and M. Al-Yami, Nonexistence results for higher order pseudo-parabolic equations in the Heisenberg group, Math. Methods Appl. Sci. 40 (2017), 1280–1287. 10.1002/mma.4052Search in Google Scholar

[19] B. Ahmad, A. Alsaedi, and M. Kirane, Blow-up of solutions to parabolic inequalities in the Heisenberg group, Electron. J. Differential Equations 2015 (2015), 167. Search in Google Scholar

[20] L. D’Ambrosio, Critical degenerate inequalities on the Heisenberg group, Manuscripta Math. 106 (2001), 519–536. 10.1007/s229-001-8031-2Search in Google Scholar

[21] M. Jleli, M. Kirane, and B. Samet, Nonexistence results for a class of evolution equations in the Heisenberg group, Fract. Calc. Appl. Anal. 18 (2015), 717–734. 10.1515/fca-2015-0044Search in Google Scholar

[22] M. Jleli, M. Kirane, and B. Samet, Nonexistence results for pseudo-parabolic equations in the Heisenberg group, Monatsh. Math. 180 (2016), 255–270. 10.1007/s00605-015-0823-7Search in Google Scholar

[23] M. Ruzhansky and N. Yessirkegenov, Existence and non-existence of global solutions for semilinear heat equations and inequalities on sub-Riemannian manifolds, and Fujita exponent on unimodular Lie groups, J. Differential Equations 308 (2022), 455–473. 10.1016/j.jde.2021.10.058Search in Google Scholar

[24] H. Brill, A semilinear Sobolev evolution equation in a Banach space, J. Differential Equations 24 (1977), 412–425. 10.1016/0022-0396(77)90009-2Search in Google Scholar

[25] C. David and W. Jet, Asymptotic behaviour of the fundamental solution to the equation of heat conduction in two temperatures, J. Math. Anal. Appl. 69 (1979), 411–418. 10.1016/0022-247X(79)90152-5Search in Google Scholar

[26] A. Dukenbayeva, Global existence and blow-up of solutions to porous medium equation for Baouendi-Grushin operator (2024), http://arXiv.org/abs/arXiv:2405.13225. 10.1515/math-2025-0144Search in Google Scholar

[27] M. Ruzhansky, B. Sabitbek, and B. Torebek, Global existence and blow-up of solutions to porous medium equation and pseudo-parabolic equation, I. Stratified group, Manuscripta Math. 171 (2023), 377–395. 10.1007/s00229-022-01390-2Search in Google Scholar

[28] S.-Y. Chung and M.-J. Choi, A new condition for the concavity method of blow-up solutions to p-Laplacian parabolic equations, J. Differential Equations 265 (2018), 6384–6399. 10.1016/j.jde.2018.07.032Search in Google Scholar

[29] G. A. Philippin and V. Proytcheva, Some remarks on the asymptotic behaviour of the solutions of a class of parabolic problems, Math. Methods Appl. Sci. 29 (2006), 297–307. 10.1002/mma.679Search in Google Scholar

[30] C. Bandle and H. Brunner, Blow-up in diffusion equations, a survey, J. Comput. Appl. Math. 97 (1998), 3–22. 10.1016/S0377-0427(98)00100-9Search in Google Scholar

[31] B. Sabitbek and B. Torebek, Global existence and blow-up of solutions to the double nonlinear porous medium equation, Discrete Continuous Dyn. Syst. Ser. B. 44 (2024), 743–767. 10.3934/dcds.2023124Search in Google Scholar

[32] B. Sabitbek, Global existence and nonexistence of solutions for semilinear wave equation with a new condition, Discrete Contin. Dyn. Syst. Ser. B 43 (2023), 2637–2657. 10.3934/dcds.2023023Search in Google Scholar

[33] C. T. Anh, P. Q. Hung, T. D. Ke, and T. T. Phong, Global attractor for a semilinear parabolic equation involving Grushin operator, Electron. J. Differential Equations 2008 (2008), 32. Search in Google Scholar

[34] M. Mihailescu, D. Stancu-Dumitru, and C. Varga, On the spectrum of a Baouendi-Grushin type operator: an Orlicz-Sobolev space setting approach, NoDEA Nonlinear Differential Equations Appl. 22 (2015), 1067–1087. 10.1007/s00030-015-0314-5Search in Google Scholar

[35] D. D. Monticelli and K. R. Payne, Maximum principles for weak solutions of degenerate elliptic equations with a uniformly elliptic direction, J. Differential Equations 247 (2009), 1993–2026. 10.1016/j.jde.2009.06.024Search in Google Scholar

[36] D. Suragan and N. Yessirkegenov, Sharp remainder of the Poincaré inequality for Baouendi-Grushin vector fields, Asian-Eur. J. Math. 16 (2023), 2350041. 10.1142/S1793557123500419Search in Google Scholar

Received: 2024-10-01

Revised: 2025-02-03

Accepted: 2025-02-23

Published Online: 2025-05-23

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/math-2025-0144

Keywords for this article

blow-up; global solution; pseudo-parabolic equation; Baouendi-Grushin operator

Creative Commons

BY 4.0