Critical Fujita exponents to a class of non-Newtonian filtration equations with fast diffusion

We consider the Cauchy problem to a class of fast-diffusion non-Newtonian filtration equations. Besides the usual degeneracy in the fast-diffusion non-Newtonian filtration, the equation is degenerate or singular at infinity, depending on the sign of the parameter related to the coefficient of diffusion. Fujita type theorems are established and the critical Fujita exponent is determined. Specially, we also prove that the nontrivial solution blows up in a finite time on the critical situation.

The purpose of this paper is to investigate the critical Fujita exponent for the following initial value problem:

where $p>1$, $0< q<1$, $\max\{-n,(n-1)/q-(n+1)\}<\mu_{1}\le\mu_{2}<p\mu_{1}+(p-1)n$, and $0\le u_{0}\in C_{0}(\mathbb {R}^{n})$.

The study of critical exponents began in 1966 by Fujita in [1], where it was proved for the initial value problem of

that the problem admits no nontrivial nonnegative global solution if $1< p< p_{c}=1+2/n$, whereas if $p>p_{c}$, it admits both global (with small data) and non-global (with large initial data) solutions. Later, in 1981, Weissler [2] proved that the critical case $p=p_{c}$ is still a blow-up case.

In Fujita’s work, the new phenomenon of nonlinear parabolic equations was discovered. From then on, there has been a lot of work on the critical Fujita exponents for various nonlinear evolution equations and systems (see, e.g., the survey papers [3, 4] and the references therein, and also [5–15]). Among those, the Fujita type theorems for the slow-diffusion non-Newtonian filtration equation

was investigated by Galaktionov in [16, 17], where $p, q>1$. He proved that $p_{c}=q+(q+1)/n$ by blow-up subsolutions and global supersolutions. Recently, the same problem for an interesting variant of (3) is studied by the authors [13]. The non-Newtonian filtration equations with fast diffusion were considered by Qi and Wang in [18], where the critical Fujita exponent was determined for the Cauchy problem of the equation

with $p>1$, $(n-1)/(n+1)< q<1$, and $\sigma>n(1-q)-q-1$. It is shown that $p_{c}=q+(q+1+\sigma)/n$ by energy functions. Obviously, they did not cover the portion $0< q\le(n-1)/(n+1)$ of the fast-diffusion range.

In the present paper, we study the problem (1), (2) and formulate the critical Fujita exponent as

and the critical situation $p=p_{c}$ is still the blow-up case. The range of m considered in this paper is $0< m<1$, the whole fast-diffusion range of (1). Like the non-Newtonian filtration equation with fast diffusion, (1) is singular at points where $\vert \nabla u\vert =0$. In addition, (1) is degenerate at $\vert x\vert =+\infty$ for $\mu_{1}>0$ and singular for $\mu_{1}<0$, different from both (3) and (4). Inspired by [11, 18, 19], to prove the solutions’ blow-up, we analyze the interaction between the nonlinear source and nonlinear diffusion via precise estimates through constructing energy functions by use of the normalized principal eigenfunction of −Δ in the unit ball $B_{1}$ of $\mathbb {R}^{n}$ with homogeneous initial-boundary condition, rather than constructing subsolutions as the author did in [16, 17]. This method for equation (1) and its special case (4) basically depends upon the nonincreasing properties in the spatial variant of solutions, which is trivial with $\mu_{1}=\mu_{2}$, while it may be invalid if $\mu_{1}<\mu_{2}$. For all these reasons, we have to overcome some technical difficulties.

This paper is arranged as follows. Some preliminaries are introduced in Section 2, including the local existence theorem, the comparison principle, and a property of solutions from propagation of disturbances. The Fujita type theorems are established in Section 3. Finally in Section 4, the critical case will be concerned.

Throughout this paper, we use $B_{r}$ to indicate the ball in $\mathbb {R}^{n}$ with radius r and center at the origin. The solution considered here is taken in the following sense.

Definition 2.1

We call

a solution to the problem (1), (2) in $(0,T)$ with $0< T\le+\infty$ if

holds for any $\phi\in C^{\infty}_{0}(\mathbb {R}^{n}\times(0,T))$ and

for any $\zeta\in C^{\infty}_{0}(\mathbb {R}^{n})$.

Like the non-Newtonian filtration equation, it is not hard to prove the well-posedness to the problem (1), (2), one can see, e.g., [20].

Next, we will prove the following proposition on a property of solutions from propagation of disturbances.

Proposition 2.1

Assume that u is a solution to the problem (1), (2) with $0\le u_{0}\in C_{0}(\mathbb {R}^{n})$ nontrivial, then $u(0,t_{0})>0$ for some $t_{0}>0$.

Proof

That $u_{0}$ is nontrivial shows that there exists $0\neq x_{0}\in\mathbb {R}^{n}$ and $\kappa, \rho>0$ such that

where $s_{+}=\max\{s,0\}$. Let

with $R(t)=\kappa^{q-1}t+\rho^{q+1}$, and $\xi>1$ independent of κ and ρ to be chosen later.

Denote

A direct calculation within D shows

Setting

then

Divide D into two sets

with $\delta>0$ satisfying

where

Then in $D^{(1)}$,

For the chosen $\delta>0$, we have in $D^{(2)}$

where

Due to

we know

So for fixed $\delta>0$ satisfying (5) and $\xi>1$ satisfying

we obtain

Clearly, the constant $\xi>1$ is independent of κ and ρ. The comparison principle implies

In particular,

with $t_{1}=\frac{\kappa^{1-q}\rho^{q+1}}{\xi}$ and

If $0\in\varGamma_{1}$, the proof is complete. Otherwise,

where

From the above argument, we have

where

with $R_{1}(t)=\kappa_{1}^{q-1}(t-t_{1})+\rho_{1}^{q+1}$. In particular,

with $t_{2}=t_{1}+\frac{\kappa_{1}^{1-q}\rho_{1}^{q+1}}{\xi}$ and

If $0\in\varGamma_{2}$, the proof is complete. Otherwise, repeat the above procedure. We get the conclusion in finite steps. □

Let us establish the Fujita type theorems.

Definition 3.1

We call u the blow-up solution to equation (1) if there exists some $0< T_{*}<+\infty$ such that

Theorem 3.1

Assume that $1< p< p_{c}=q+(q+1+\mu_{2})/(n+\mu_{1})$ and $0\le u_{0}\in C_{0}(\mathbb {R}^{n})$ is nontrivial. Then the problem (1), (2) admits a blow-up solution.

Proof

Due to Proposition 2.1, we may assume $u_{0}(0)>0$. By the comparison principle, we only need to prove the conclusion for radial and nonincreasing $u_{0}(x)$, i.e.,

where $h_{0}\in C^{1}_{0}([0,+\infty))$ satisfies $h'_{0}(0)=0$ and $h'_{0}(r)\le0$ for $r>0$. With such initial data, the solution u is also radial, namely

If $\mu_{1}=\mu_{2}$, it is easy to know that u is also nonincreasing by a standard regularization argument and the maximum principle. However, this method is invalid if $\mu_{1}<\mu_{2}$. In the following discussion, we will first of all consider a nonincreasing u, namely $h(r,t)$ is nonincreasing with respect to $r\in[0,+\infty)$ for any $t\ge0$, and we treat the general case finally.

Let

where f is the principal eigenfunction of −Δ in the unit ball $B_{1}$ of $\mathbb {R}^{n}$ with homogeneous initial-boundary condition, normalized by $\Vert f\Vert _{L^{\infty}(B_{1})}=1$. For $l>1$, define

Then

with $M_{0}>0$ independent of l. Set

Definition 2.1 gives

For radial and nonincreasing $u(x,t)$, one has

and

where ν is the unit outer normal to $\partial B_{2l}$. Hence

The Hölder inequality yields

with $M_{1}>0$ independent of l, which, together with (8), implies

By the Hölder inequality,

and hence

with $M_{2}>0$ independent of l. Equations (9) and (10) show that

We mention that the above discussion holds provided that $p>1$.

If $p< p_{c}=q+(q+1+\mu_{2})/(n+\mu_{1})$, then

Notice that $\eta_{l}$ is nondecreasing with respect to $l\in(1,+\infty)$ and $\sup\{\eta_{l}(0):l\in(1,+\infty)\}>0$, and from (11) one shows that, for $l>1$ large enough, there exists a constant $\delta>0$ such that

So there exists some $0< T_{*}<+\infty$ such that

Due to $\operatorname{supp} \psi_{l}=B_{2l}$, we obtain

Next, for the general case without the assumption that $u(x,t)$ is nonincreasing, define

Then $\underline{u}(x,t)$ is nonincreasing,

and

From the above argument, we get

for some $0<\tilde{T}_{*}<+\infty$, and (13) ensures that u is a blow-up solution. □

Let us turn to the case $p>p_{c}$. Suppose that

where

is a self-similar solution to (1). It is easy to show that $V(r)$ solves

Lemma 3.1

Assume that $p>p_{c}=q+(q+1+\mu_{2})/(n+\mu_{1})$. Then for $\varepsilon>0$ small enough, the function

where

is a supersolution to equation (17), i.e.

Proof

It is not hard to show that it suffices to verify

namely

From the definition of $[\varepsilon\gamma\lambda\rho(\varepsilon )]^{q}=\beta\varepsilon$, we have

Due to $p>p_{c}=q+(q+1+\mu_{2})/(n+\mu_{1})$ and $(n-1)/q-(n+1)<\mu_{1}\le\mu_{2}$,

Hence,

holds for sufficiently small $\varepsilon>0$, and (18) is obtained. □

Theorem 3.2

Assume that $p>p_{c}=q+(q+1+\mu_{2})/(n+\mu_{1})$. Then the solution to the problem (1), (2) exists globally with small $u_{0}$, or blows up with large $u_{0}$.

Proof

Let $V(r)$ be a supersolution to equation (17) in Lemma 3.1. From

one can show that $U(x,t)$ given in (15) is a supersolution to equation (1) with α and β given in (16). Therefore, the comparison principle implies the problem (1), (2) has a nontrivial global solution with small $u_{0}$.

Let us turn to the case of large $u_{0}$. Denote by the radial u a solution to the problem (1), (2). Temporarily suppose u is nonincreasing. Then (11) holds with $\eta_{l}$ defined by (7). If $u_{0}$ is so large that

holds for some $l_{0}>1$, then from (11), we get

and

for some $\delta_{0}>0$. Therefore, u is a blow-up solution.

For the general case without the assumption that $u(x,t)$ is nonincreasing, considering a new function just as in the proof of Theorem 3.1, one can also see that u is a blow-up solution. □

Now, let us deal with the critical case $p=p_{c}=q+(q+1+\mu_{2})/(n+\mu_{1})$. Let $\psi_{l}$, $\eta_{l}$, $M_{1}$, $M_{2}$ be defined as in the previous section.

Lemma 4.1

Assume that $p=p_{c}=q+(q+1+\mu_{2})/(n+\mu_{1})$ and u is a nontrivial, global, radial, and nonincreasing solution to the problem (1), (2). Then

holds for some constant $M>0$ independent of t.

Proof

$p=p_{c}$ yields

Then, for the global, radial and nonincreasing solution u, from (11), we have

Otherwise, u blows up in a finite time. Therefore (19) holds for some constant $M>0$ owing to

 □

Lemma 4.2

Assume that $p=p_{c}=q+(q+1+\mu_{2})/(n+\mu_{1})$, u be a nontrivial, radial, and nonincreasing solution to the problem (1), (2) and $0<\theta<1$. Then

holds for any $l>1$ with a constant $M_{3}>0$ independent of l.

Proof

For any $l>1$, the Hölder inequality yields

with $M>0$ a constant independent of l, which, together with (10) and (8), implies

Then (20) holds due to

 □

Lemma 4.3

Assume that $p=p_{c}=q+(q+1+\mu_{2})/(n+\mu_{1})$, u be a nontrivial, radial and nonincreasing solution to the problem (1), (2). Then

holds for some constant $M_{4}>0$ independent of l.

Proof

From (9), we have

Then the Young inequality gives

 □

We are ready to prove the blow-up theorem of Fujita type for the critical case $p=p_{c}=q+(q+1+\mu_{2})/(n+\mu_{1})$.

Theorem 4.1

Assume that $p=p_{c}=q+(q+1+\mu_{2})/(n+\mu_{1})$ and u is a solution to the problem (1), (2) with $0\le u_{0}\in C_{0}(\mathbb {R}^{n})$ nontrivial. Then the problem (1), (2) admits a blow-up solution.

Proof

Similarly to the proof of Theorem 3.1, at first assume $u_{0}$ is radial and nonincreasing. Then u is radial, given by (6). Denote

From (19) and the nontriviality of u, $0<\varLambda<+\infty$. For any $0<\sigma<\varLambda$, due to (22) and $\eta_{l}$ being nondecreasing with respect to $l\in(1,+\infty)$, there exist $\omega_{0}\ge0$ and $l_{0}>2$ such that

Then it follows from (21) that

Thus

Choosing $l=l_{0}$ in (20), we have

Fix $\sigma_{0}\in(0,\varLambda)$ and $M_{5}>0$, independent of $l_{0}$, such that

Then

where

Hence

Notice

one gets

with a positive constant

independent of $l_{0}$. Similarly, we reason

The same argument yields

with

Repeating the procedure, one can show that

with

Therefore

which contradicts (19).

Now, for the general case without the assumption that $u(x,t)$ is nonincreasing, consider $\underline{u}(x,t)$ defined by (12), which is nonincreasing and satisfies (13) and (14). Therefore, the conclusions of Lemmas 4.1-4.3 are all valid for $\underline{u}$. Similar to the above argument, one can show that $\underline{u}$ blows up in some $0< T_{*}<+\infty$, and thus u is a blow-up solution. □

This work is supported by the National Natural Science Foundation of China (Grant Nos. 11222106 and 11571137).

Affiliations

1. School of Mathematics, Jilin University, Changchun, 130012, China

   • Mingjun Zhou
   •  & Huilai Li

2. School of Mathematics and Statistics, Beihua University, Jilin, 132013, China

   • Wei Guo

3. College of Computer Science and Technology, Jilin University, Changchun, 130012, China

   • Xu Zhou

Corresponding author

Correspondence to Wei Guo.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All the authors contributed to each part of this study equally and approved the final version of the manuscript.

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