Fujita-type theorems for a quasilinear parabolic differential inequality with weighted nonlocal source term
-
Yuepeng Li
Abstract
This work is concerned with the nonexistence of nontrivial nonnegative weak solutions for a quasilinear parabolic differential inequality with weighted nonlocal source term in the whole space, which involves weighted polytropic filtration operator or generalized mean curvature operator. We establish the new critical Fujita exponents containing the first and second types. The key ingredient of the technique in proof is the test function method developed by Mitidieri and Pohozaev. No use of comparison and maximum principles or assumptions on symmetry or behavior at infinity of the solutions are required.
1 Introduction
We consider a quasilinear parabolic differential inequality with weighted nonlocal source
where
which may be singular or degenerate at the origin. Note that the differential inequality (1.1) is called nonlocal differential inequality in the sense that it is not defined pointwise. Meanwhile, norm type nonlocal terms appear in population dynamics and theory of biological populations, see [7,22].
We are interested in finding sufficient conditions for the nonexistence of nonnegative nontrivial weak solutions for (1.1), especially the results related to critical Fujita exponents. The pioneering article in this subject was by Fujita [6], where he considered the Cauchy problem for semilinear parabolic equation
Motivated by previous results of Mitidieri and Pohozaev in [17,18] and Kartsatos and Kurta in [12], we first study the Fujita-type nonexistence of nonnegative nontrivial weak solutions for a wide class of nonlocal quasilinear parabolic differential inequalities in the framework of the test function. No use of comparison and maximum principles or assumptions on symmetry or behavior at infinity of the solutions are required.
We assume throughout this article that
where
for all
where
For the case
The general class of weakly coercive operators satisfying the structural condition (1.5) was introduced by Farina and Serrin in [4,5] for studying the ellipticproblem, while the subcase of (1.5) with
Operators satisfying the aforementioned assumptions can be given by
or
Thus, our analysis includes weighted polytropic filtration operator and generalized mean curvature operator. In particular, the aforementioned generalized mean curvature operator satisfies (1.5) for all
We point out that the presence of various components such as the singular coefficient in the operator on the left-hand side of (1.1) and singular coefficient, weighted norm type nonlocal term, power like source, etc. on the right-hand side makes the study of Fujita-type nonexistence results very delicate. In particular, the derivation of a priori estimates is much more complicated because of the fact that several factors appear.
The main results are stated as follows.
Theorem 1.1
Let
If
(1.7)then (1.1) does not admit nontrivial nonnegative weak solutions in
for
If both q and r are not equal to 1, and
then (1.1) does not admit nontrivial nonnegative weak solutions in
Theorem 1.2
Let
If
(1.9)then (1.1) does not admit nontrivial positive weak solutions in
for
If both q and r are not equal to 1, and
then (1.1) does not admit nontrivial positive weak solutions in
Remark 1.1
Conditions (1.6) and (1.8) force that (1.7) and (1.9) are not empty intervals, respectively.
As a consequence of our main Theorems 1.1 and 1.2, by using the polytropic filtration type operator in problem (1.1) and rewriting it as
where
Corollary 1.1
Let
If
then (1.10) has no nontrivial positive (or nonnegative) weak solution in
for
If both q and r are not equal to 1, and
then (1.10) has no nontrivial positive (or nonnegative) weak solution in
for
Corollary 1.1 implies the critical Fujita exponent for the power of the source at the right-hand side of inequality (1.10) is
The analogous of Corollary 1.1 holds for the mean curvature case, which can be considered morally for the case
and rewrite (1.1) as
where
Corollary 1.2
Let
If
then (1.11) has no nontrivial positive (or nonnegative) weak solution in
for
If both q and r are not equal to 1 and
then (1.11) has no nontrivial positive (or nonnegative) solution in
for
From Corollary 1.2, the critical Fujita exponent for the power
In conclusion, comparison of the main results on the first critical Fujita exponents obtained earlier with the existing results is presented in Table 1.
First critical Fujita exponents
| Operators in (1.1) | First critical Fujita exponents for
|
Comparison with existing results |
|---|---|---|
|
|
|
New |
| Polytropic filtration operator
|
|
|
| Mean curvature operator
|
|
|
On the other hand, it is obvious from [10,25] that the classical critical Fujita exponent
Theorem 1.3
Let
If there exists
and
then (1.1) does not admit nonnegative nontrivial solutions belonging to the class
Theorem 1.4
Let
If there exists
then (1.1) does not admit nontrivial positive solutions belonging to the class
Remark 1.2
It can be compared between the first critical Fujita exponents given in Theorems 1.1 and 1.2 and the second critical exponents given in Theorems 1.3 and 1.4 under appropriate conditions. Actually, if
or
we have
In addition, we give the second critical Fujita exponent results corresponding with Corollaries 1.1 and 1.2.
Corollary 1.3
Let
If there exists
then (1.10) has no nontrivial positive (or nonnegative) weak solution belonging to the class
for
We obtain a new second critical exponent
Corollary 1.4
Let
If there exists
then (1.11) has no nontrivial positive (or nonnegative) weak solution belonging to the class
for
Remark 1.3
Similar to Remark 1.2, the first critical exponents in Corollaries 1.1 and 1.2 can be compared with the second ones in Corollaries 1.3 and 1.4 under appropriate conditions. By taking
Indeed, the second critical Fujita exponent
To sum up, we compare the main results on the second critical Fujita exponents with the existing results and obtain Table 2.
Second critical Fujita exponents
| Operators in (1.1) | Second critical Fujita exponents for
|
Comparison with existing results |
|---|---|---|
|
|
|
New |
| Polytropic filtration operator
|
|
|
| Mean curvature operator | ||
|
|
|
New |
This article is organized as follows. In Section 2, we introduce the preparatory knowledge. In Section 3, we present several technical lemmas and obtain fine a priori estimates. Finally, we present the detailed proofs of main results in Section 4.
2 Preliminaries
In this section, we introduce some notations, definitions, and the careful selections of test functions.
Throughout this article, we denote
First, we define the weak solution of (1.1).
Definition 2.1
For a weak solution of (1.1), we mean a nonnegative function
Obviously, when the equal sign holds in (1.1), we can consider test functions
When necessary, we make use of the following weak formulation of (1.1)
for any nonnegative test function
Next, we construct the test function in the form of separated variables with parameters carefully. Let
Moreover,
where
For the space variable, we take
thus,
where
For the time variable, consider
with
Now, we define, for
Obviously
Then, we give the test functions in two cases.
Case 1. For
where
Here,
Clearly for all
Case 2. For
with
Similarly,
3 A priori estimates
In this section, we present a priori estimates of the solution needed in the proofs of the main results. To this end, we will give several technical preliminaries in Subsection 3.1 and derive the corresponding a priori estimates in two cases in Subsection 3.2.
3.1 Technical lemmas
In this subsection, we prepare some technical preparatory lemmas in order to obtain a priori estimates.
Lemma 3.1
Let
where
and
Proof
Rewrite the integrand in the integral on the left-hand side of (3.1) as
We take
It follows that
We claim
Due to
Thus, by Hölder inequality with exponents
By substituting it into the left-hand side of (3.1), we have
Then, by Young’s inequality with exponents
which completes the proof of Lemma 3.1.□
Lemma 3.2
Let
If
where
and
Proof
The integrand in the integral on the left-hand side of (3.2) can be rewritten as follows:
Let
It implies that
We claim
It follows from
Hence, by using Hölder inequality with exponents
By inserting the aforementioned formula into the left-hand side of (3.2), we have
Then, by applying Young’s inequality with exponents
Lemma 3.2 is proved.□
Lemma 3.3
Let
If
where
Proof
We rewrite the integrand in the integral on the left-hand side of (3.3) as follows:
Choose
It follows that
We claim
Then,
which gives
So we can use Hölder inequality with exponents
By substituting it into the left-hand side of (3.3), we have
Then, by applying Hölder inequality again with exponents
which yields (3.3).□
Lemma 3.4
Let
where
Proof
Rewrite the integrand in the integral on the left-hand side of (3.4) as follows:
We take
By simple calculations, we have
We claim
Since
Thus, by Hölder inequality with exponents
By substituting it into the left-hand side of (3.4), we have
Then, by applying again Hölder inequality with exponents
Lemma 3.4 is proved.□
3.2 A priori estimates
In this subsection, by using the preparatory lemmas obtained in Subsection 3.1, we give fine a priori estimates for the two cases
Proposition 3.1
Let
Assume
where
Proof
Let
Since
Being
and (3.8) gives
Since
From (1.5), we have for all
Further, by Young’s inequality with exponents
Since
Since all the exponents of
Then, we estimate the last two terms by Lemmas 3.1 and 3.2 with
which gives, since
We next estimate the last two integrals in (3.11). The exponents of
By the definition of
Thus, we have for the first one that
On the basis of the scale transformation
and
In a similar way, we estimate the last integral in (3.11)
Choosing
Under some extra assumptions on the parameters, we obtain a refined a priori estimate for the critical case when
Proposition 3.2
Let
For
where
and here
Proof
By choosing
We take
By applying Hölder inequality with exponents
By combining it with Lemmas 3.3, 3.4, and (3.6), we have
Since
and inequality (3.12) is proved.□
Similar to Proposition 3.1, we derive a priori estimate of the positive solution to the differential inequality (1.1).
Proposition 3.3
Let
Assume
where
Proof
The proof is analogous to that of Proposition 3.1 where we deal with positive solutions and take the test function as
By applying Lemmas 3.1 and 3.2 with
We choose
and
Taking
The analogous of Proposition 3.2 holds when
Proposition 3.4
Let
Assume
where
and here,
Proof
The proof here is similar to that of Proposition 3.2, but now we deal with the positive solution since
By substituting (3.15) into the second term on the right-hand side and applying Lemmas 3.4 and 3.3 with
By taking
inequality (3.18) is proved.□
4 Proofs of main theorems
In this section, we show the process of the proofs for the main Theorems 1.1–1.4 in detail.
4.1 First critical Fujita exponent
This subsection is devoted to prove Theorems 1.1 and 1.2 on the first critical Fujita exponents.
Proof of Theorem 1.1
By
which follows, since
To make it easier to discuss the exponents of
where
and
where
Thanks to
for
which is equivalent to
Condition
and we have
We claim that when
Indeed,
from
Due to
In the following, we discuss the exponents of
Case 1. If
which implies
Case 2. If
By using Proposition 3.2, we obtain another a priori estimate, namely, (3.12)
Taking
Thus, we obtain from (3.12) that
Since
Then, let
By the absolute continuity of the Lebesgue integral ([23, p. 20]), let
Now, by passing to the limit as
which completes the proof of the theorem.□
Proof of Theorem 1.2
The proof is exactly the same of Theorem 1.1 but now
where
and
Now we solve equation
Due to
A straightforward computation shows us that
By arguing as in the proof of Theorem 1.1, we obtain, if
then
Next, we discuss the exponents of
Case 1. If
Case 2. If
By (3.18) in Proposition 3.4, we obtain
Then, similar to the proof of Theorem 1.1, we derive
which completes the proof of the theorem.□
4.2 Second critical Fujita exponent
This subsection shows the proofs of Theorems 1.3 and 1.4 related to the second critical Fujita exponents.
Proof of Theorem 1.3
By
By (1.12), we have
where
For
One can see that the expression on the right-hand side in (4.6) arrives its minimum at
By substituting it into (4.6), we obtain
Meanwhile, by (1.13),
Proof of Theorem 1.4
The proof is analogous to that of Theorem 1.3. Suppose
where
Taking
The expression on the right-hand side in (4.8) arrives its minimum at
By substituting it into (4.8), we obtain
By (1.14), let
Acknowledgment
The authors would like to deeply thank all the reviewers for their insightful and constructive comments.
-
Funding information: The work was supported by the Natural Science Foundation of Shandong Province of China (No. ZR2019MA072).
-
Author contributions: All authors contributed equally to the manuscript and read and approved the final manuscript.
-
Conflict of interest: The authors declare that there is no conflict of interests regarding the publication of this article.
-
Data availability statement: Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.
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- Infinitely many localized semiclassical states for nonlinear Kirchhoff-type equation
- Fujita-type theorems for a quasilinear parabolic differential inequality with weighted nonlocal source term
- Approximations of center manifolds for delay stochastic differential equations with additive noise
- Periodic solutions to a class of distributed delay differential equations via variational methods
- Groundstate for the Schrödinger-Poisson-Slater equation involving the Coulomb-Sobolev critical exponent
- Existence of nontrivial solutions for the Klein-Gordon-Maxwell system with Berestycki-Lions conditions
- Global Sobolev regular solution for Boussinesq system
- Normalized solutions for the p-Laplacian equation with a trapping potential
- Nonlinear elliptic–parabolic problem involving p-Dirichlet-to-Neumann operator with critical exponent
- Blow-up for compressible Euler system with space-dependent damping in 1-D
- High energy solutions of general Kirchhoff type equations without the Ambrosetti-Rabinowitz type condition
- On the dynamics of grounded shallow ice sheets: Modeling and analysis
- A survey on some vanishing viscosity limit results
- Blow-up for logarithmic viscoelastic equations with delay and acoustic boundary conditions
- Generalized Liouville theorem for viscosity solutions to a singular Monge-Ampère equation
- Front propagation in a double degenerate equation with delay
- Positive solutions for a class of singular (p, q)-equations
- Higher integrability for anisotropic parabolic systems of p-Laplace type
- The Poincaré map of degenerate monodromic singularities with Puiseux inverse integrating factor
- On a system of multi-component Ginzburg-Landau vortices
- Existence and concentration of solutions to Kirchhoff-type equations in ℝ2 with steep potential well vanishing at infinity and exponential critical nonlinearities
- Multiplicity results for fractional Schrödinger-Kirchhoff systems involving critical nonlinearities
- On double phase Kirchhoff problems with singular nonlinearity
- Estimates for eigenvalues of the Neumann and Steklov problems
- Nodal solutions with a prescribed number of nodes for the Kirchhoff-type problem with an asymptotically cubic term
- Dirichlet problems involving the Hardy-Leray operators with multiple polars
- Incompressible limit for compressible viscoelastic flows with large velocity
- Characterizations for the genuine Calderón-Zygmund operators and commutators on generalized Orlicz-Morrey spaces
- Non-local gradients in bounded domains motivated by continuum mechanics: Fundamental theorem of calculus and embeddings
- Noncoercive parabolic obstacle problems
- Touchdown solutions in general MEMS models
- Existence and nonexistence of nontrivial solutions for the Schrödinger-Poisson system with zero mass potential
- Short-time existence of a quasi-stationary fluid–structure interaction problem for plaque growth
- Fine bounds for best constants of fractional subcritical Sobolev embeddings and applications to nonlocal PDEs
- Symmetries of Ricci flows
- Asymptotic behavior of solutions of a second-order nonlinear discrete equation of Emden-Fowler type
- On the topological gradient method for an inverse problem resolution
- Supersolutions to nonautonomous Choquard equations in general domains
- Uniform complex time heat Kernel estimates without Gaussian bounds
- Global existence for time-dependent damped wave equations with nonlinear memory
- Normalized solutions for a critical fractional Choquard equation with a nonlocal perturbation
- Reversed Hardy-Littlewood-Sobolev inequalities with weights on the Heisenberg group
- Lamé system with weak damping and nonlinear time-varying delay
- Global well posedness for the semilinear edge-degenerate parabolic equations on singular manifolds
- Blowup in L1(Ω)-norm and global existence for time-fractional diffusion equations with polynomial semilinear terms
- Boundary regularity results for minimisers of convex functionals with (p, q)-growth
- Parametric singular double phase Dirichlet problems
- Special Issue on Nonlinear analysis: Perspectives and synergies
- Editorial to Special issue “Nonlinear analysis: Perspectives and synergies”
- Identification of discontinuous parameters in double phase obstacle problems
- Global boundedness to a 3D chemotaxis-Stokes system with porous medium cell diffusion and general sensitivity
- On regular solutions to compressible radiation hydrodynamic equations with far field vacuum
- On Cauchy problem for fractional parabolic-elliptic Keller-Segel model
- The evolution of immersed locally convex plane curves driven by anisotropic curvature flow
- Stability of combination of rarefaction waves with viscous contact wave for compressible Navier-Stokes equations with temperature-dependent transport coefficients and large data
- On concave perturbations of a periodic elliptic problem in R2 involving critical exponential growth
- Existence of solutions for a double-phase variable exponent equation without the Ambrosetti-Rabinowitz condition
