Abstract
This paper deals with the existence and non-existence of the global solutions to the Cauchy problem of a semilinear parabolic equation with a gradient term. The blow-up theorems of Fujita type are established and the critical Fujita exponent is determined by the behavior of the three variable coefficients at infinity associated to the gradient term and the diffusion–reaction terms, respectively, as well as the spacial dimension.
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1 Introduction
In the paper, we investigate the blow-up theorems of Fujita type for the following Cauchy problem:
where \(p>1\), \(-2<\lambda_{1}\leq\lambda_{2}\), \(0\leq u_{0}\in C_{0}(\mathbb {R}^{n})\) and \(b\in C^{1}([0,+\infty))\) satisfies
and in the case that \(-n-\lambda_{1}<\kappa\leq+\infty\), b also satisfies
The critical exponents for nonlinear diffusion equations have attached extensive attention since 1966, when Fujita [1] proved that, for the Cauchy problem of Eq. (1.1) with \(b\equiv0\) and \(\lambda_{1}=\lambda_{2}=0\), the nontrivial nonnegative solution blows up in a finite time if \(1< p< p_{c}=1+2/n\), whereas it exists globally for small initial data and blows up in a finite time for large ones if \(p>p_{c}=1+2/n\). This result reveals that the exponent p of the nonlinear reaction plays a remarkable role in affecting the properties of solutions. We call \(p_{c}\) with the above properties the critical Fujita exponent and the similar result a blow-up theorem of Fujita type. There have been many kinds of extensions of Fujita’s results since then, such as different types of parabolic equations and systems with or without degeneracies or singularities, various geometries of domains, different nonlinear reactions or nonhomogeneous boundary sources, etc. One can see the survey papers [2, 3] and the references therein, and more recent work [4–18]. For the Cauchy problem of
with \(\mathbf{b}_{0}\) being a nonzero constant vector, Aguirre and Escobedo [19] showed that
is its critical Fujita exponent. Wang and Zheng [11] considered the Cauchy problem of Eq. (1.1) with \(b\equiv0\), and showed that the critical Fujita exponent is
Recently, the Cauchy problem of Eq. (1.1) with \(\lambda _{1}=\lambda_{2}=0\) was studied in [18] and it was shown that
As to Neumann exterior problems, Levine and Zhang [20] investigated the critical Fujita exponent of the homogeneous Neumann exterior problem of (1.1) with \(b\equiv0\) and \(\lambda_{1}=\lambda_{2}=0\), and proved that \(p_{c}\) is still \(1+2/n\). In [21], Zheng and Wang concerned the homogeneous Neumann exterior problem of (1.1) with
and formulated the critical Fujita exponent as
Moreover, the general case of b is considered in [8] if \(0\le \lambda_{1}\leq\lambda_{2}\le p\lambda_{1}+(p-1)n\) and \(\kappa\ge0\).
In this paper, we investigate the blow-up theorems of Fujita type for the Cauchy problem (1.1), (1.2). It is proved that the critical Fujita exponent to the problem is
That is to say, if \(1< p< p_{c}\), there does not exist any nontrivial nonnegative global solution, whereas if \(p>p_{c}\), there exist both nontrivial nonnegative global and blow-up solutions. The technique used in this paper is mainly inspired by [11, 18, 21, 22]. To prove the blow-up of solutions, we use precise energy integral estimates instead of constructing subsolutions. For the global existence of nontrivial solutions, we construct a nontrivial global supersolution. It should be noted that we have to seek a complicated supersolution and do some precise calculations in order to overcome the difficulty from the non-self-similar construction of (1.1). Furthermore, the properties of such models which will be proved in the paper provide theoretical foundation for the numerical simulation which involved difference schemes.
The paper is organized as follows. Some preliminaries and main results are introduced in Sect. 2, such as the local well-posedness of the problem (1.1), (1.2) and some auxiliary lemmas to be used later, as well as the blow-up theorems of Fujita type. The main results are proved in Sect. 3.
2 Preliminaries and main results
The solutions to the problem (1.1), (1.2) are defined as follows.
Definition 2.1
A nonnegative function u is called a solution to the problem (1.1), (1.2) in \((0,T)\) with \(0< T\le+\infty\), if
and
Definition 2.2
A solution u to the problem (1.1), (1.2) is called a blow-up solution if there exists some \(T_{*}\in(0,+\infty)\), which is called blow-up time, such that
Otherwise, u is called a global solution.
For \(0\le u_{0}\in C_{0}(\mathbb {R}^{n})\) and \(b\in C^{1}([0,+\infty))\) and \(p>1\), one can establish the existence, uniqueness and the comparison principle for solutions to the problem (1.1), (1.2) locally in time by use of the classical theory on parabolic equations (see, e.g., [23]).
The blow-up theorems of Fujita type for the problem (1.1), (1.2) are stated as follows.
Theorem 2.1
Assume that \(b\in C^{1}([0,+\infty))\) satisfies (1.3) and (1.4) with \(-\infty\le\kappa<+\infty\). If \(1< p< p_{c}\) with \(p_{c}\) given by (1.5), then, for any nontrivial \(0\le u_{0}\in C_{0}(\mathbb {R}^{n})\), the solution to the problem (1.1), (1.2) must blow up in a finite time.
Theorem 2.2
Assume that \(b\in C^{1}([0,+\infty))\) satisfies (1.3) and (1.4) with \(-n<\kappa\le+\infty\). If \(p>p_{c}\) with \(p_{c}\) given by (1.5), then there exist both nontrivial nonnegative global and blow-up solutions to the problem (1.1), (1.2).
3 Proofs of main results
To prove Theorem 2.1, the following auxiliary lemma is needed. We omit the proof and a similar one may be found in [18, 21].
Lemma 3.1
Assume that \(b\in C^{1}([0,+\infty))\) satisfies (1.3) and (1.4) with \(-\infty\le\kappa<+\infty\), u is a solution to the problem (1.1), (1.2), and
with
Then there exist three numbers \(R_{0}>0\), \(\delta>1\) and \(M_{0}>0\) depending only on n and b, such that, for any \(R>R_{0}\),
in the distribution sense, where \(B_{r}\) denotes the open ball in \(\mathbb {R}^{n}\) with radius r and centered at the origin.
Remark 3.1
For the case \(\kappa=+\infty\), one can prove that (3.1) holds for each fixed \(R>0\), but \(\delta>1\) and \(M_{0}>0\) depend also on R.
Proof of Theorem 2.1
Let \(\eta_{R}\), h, \(R_{0}\), δ and \(M_{0}\) be introduced in Lemma 3.1. It follows from \(-\infty\le\kappa<+\infty\) and \(1< p< p_{c}\) that
Fix \(\tilde{\kappa}>\kappa\) to satisfy
which, together with (1.3), shows that there exists \(R_{1}>1\) such that
For any \(R>R_{1}\), one can get
and
where
Therefore,
where \(\chi_{[0,\delta R]}\) is the characteristic function of the interval \([0,\delta R]\), while \(K>0\) depends only on n, b, \(R_{1}\), δ and κ̃. Let u be the solution to the problem (1.1), (1.2), and denote
For any \(R>\max\{R_{0}, R_{1}\}\), Lemma 3.1 implies
The Hölder inequality and (3.3) yield
where \(\omega_{n}\) is the volume of the unit ball in \(\mathbb {R}^{n}\), while \(M_{1}>0\) depends only on n, b, \(R_{1}\), δ and κ̃. Substituting (3.5) into (3.4) gives
It follows from (3.2), (3.3) and the Hölder inequality that
with \(M_{2}>0\) depending only on n, b, \(R_{1}\), δ and κ̃, and
Substituting (3.7) into (3.6), one gets, for any \(R>\max\{R_{0},R_{1}\}\),
Note that (3.2) implies
while \(w_{R}(0)\) is nondecreasing with respect to \(R\in(0,+\infty)\) and
Therefore, there exists \(R_{2}>0\) such that, for any \(R>R_{2}\),
Fix \(R>\max\{R_{0},R_{1},R_{2}\}\). (3.8) and (3.9) yield
Since \(p>1\), there exists \(T_{*}>0\) such that
It follows from \(\operatorname{supp}\eta_{R}( \vert x \vert )=\overline{B}_{\delta R}\) that
i.e., u blows up in a finite time. □
Now, let us prove Theorem 2.2. Firstly, we study self-similar supersolutions of (1.1) of the form
with
and \(\tau>0\) will be determined. If \(U\in C^{1,1}([0,+\infty))\) with \(U'\leq0\) in \((0,+\infty)\) satisfies
then u given by (3.10) is a supersolution to (1.1).
Lemma 3.2
Assume that \(b\in C^{1}([0,+\infty))\) satisfies (1.3) and (1.4) with \(-n-\lambda_{1}<\kappa\leq+\infty\), \(p>p_{c}\),
with \(A\in C^{1,1}([0,+\infty))\) satisfies \(A(0)=0\) and
where \(0< l<1\) will be determined,
with \(\kappa_{1}\), \(\kappa_{2}\) satisfying
Then there exist \(\varepsilon>0\), \(0< l<1\) and \(\tau>0\) such that u given by (3.10) and (3.11) is a supersolution to (1.1).
Proof
The choice of \(\kappa_{1}\), \(\kappa_{2}\) leads to \(A_{2}<\beta<A_{1}\). Fix
Additionally, (1.3) allows us to choose \(\tau>0\) sufficiently large such that
For \(0< r< l^{2}\) and \(t>0\), we have from (1.3) and (3.12)
Then, for \(l^{2}< r< l\) and \(t>0\), (3.12) and (3.13) ensure that
Finally, for \(r>l\) and \(t>0\), (3.12) and (3.13) guarantee that
Due to \(-2<\lambda_{1}\leq\lambda_{2}\), \(p>1\) and the definition of the function \(A(r)\),
Let \(\varepsilon>0\) be sufficiently small such that
Therefore, u given by (3.10) and (3.11) is a supersolution to (1.1). □
Proof of Theorem 2.2
The comparison principle and Lemma 3.2 show that there exists a nontrivial global solution to the problem (1.1), (1.2). We will show that the problem also admits a blow-up solutions. Fix \(R>R_{0}\). Assume that u is a solution to the problem (1.1), (1.2). Lemma 3.1 and Remark 3.1 imply that
where \(\eta_{R}\), \(R_{0}\), δ, \(M_{0}\) and \(w_{R}(t)\) are given in Lemma 3.1, Remark 3.1 and the proof of Theorem 2.1. The Hölder inequality yields
which implies
Substituting (3.18) into (3.17) we get
If \(u_{0}\) is so large that
then (3.19) leads to
The same argument as in the proof of Theorem 2.1 shows that u must blow up in a finite time. □
Remark 3.2
For the critical case \(p=p_{c}\) with \(-n-\lambda_{1}<\kappa<+\infty\). we need an additional condition (see [18]) that
Similar to the proof in critical case in [18, 21], one can show the blow-up of the solutions to the problem (1.1), (1.2) for the critical case \(p=p_{c}\) with \(-n-\lambda _{1}<\kappa<+\infty\) if (3.20) holds.
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This work is supported by the National Natural Science Foundation of China (11571137 and 11601182).
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Na, Y., Zhou, M., Zhou, X. et al. Blow-up theorems of Fujita type for a semilinear parabolic equation with a gradient term. Adv Differ Equ 2018, 128 (2018). https://doi.org/10.1186/s13662-018-1582-2
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DOI: https://doi.org/10.1186/s13662-018-1582-2