Multiple studies have addressed the blowup time of the Fujitatype equation. However, an explicit and sharp inclusion method that tackles this problem is still missing due to several challenging issues. In this paper, we propose a method for obtaining a computable and mathematically rigorous inclusion of the \(L^2(\varOmega )\) blowup time of a solution to the Fujitatype equation subject to initial and Dirichlet boundary conditions using a numerical verification method. More specifically, we develop a computerassisted method, by using the numerically verified solution for nonlinear parabolic equations and its estimation of the energy functional, which proves that the concerned solution blows up in the \(L^2(\varOmega )\) sense in finite time with a rigorous estimation of this time. To illustrate how our method actually works, we consider the Fujitatype equation with Dirichlet boundary conditions and the initial function \(u(0,x)=\frac{192}{5}x(x1)(x^2x1)\) in a onedimensional domain \(\varOmega\) and demonstrate its efficiency in predicting \(L^2(\varOmega )\) blowup time. The existing theory cannot prove that the solution of the equation blows up in \(L^2(\varOmega )\). However, our proposed method shows that the solution is the \(L^2(\varOmega )\) blowup solution and the \(L^2(\varOmega )\) blowup time is in the interval (0.3068, 0.317713].
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1 Introduction
Let \(\varOmega \subset \mathbb {R}^N(N\in \mathbb {N})\) be a bounded Lipschitz domain. In this paper, we propose a numerical verification algorithm for obtaining an explicit and computable inclusion (an interval containing the exact value) of the blowup time of solutions to the Fujitatype equation:
For \(1\le q<\infty\), \(L^q(\varOmega )\) is the set of the qth power Lebesgue integrable realvalued functions in \(\varOmega\) endowed with the norm \(\Vert u\Vert _{L^{q}(\varOmega )}:=\left( \int _{\varOmega }u(x)^q dx\right) ^{\frac{1}{q}}\). The \(L^2(\varOmega )\) inner product is defined as \((u,v)_{L^2(\varOmega )}:=\int _{\varOmega }u(x)v(x)dx\) for \(u,v\in L^2(\varOmega )\). The \(H^1_0(\varOmega )\) is the set of once weakly differentiable \(L^2(\varOmega )\) functions vanishing on the boundary of \(\varOmega\). The inner product and the norm of \(H^1_0(\varOmega )\) are defined as \((u,v)_{H^1_0(\varOmega )}:=(\nabla u,\nabla v)_{L^2(\varOmega )}\) and \(\Vert u\Vert _{H^{1}_{0}(\varOmega )}:=\sqrt{(u,u)_{H^1_0(\varOmega )}}\) for \(u, v\in H^1_0(\varOmega )\), respectively. Let \(H^{1}(\varOmega )\) be the dual space of \(H^1_0(\varOmega )\) and let \(\langle \cdot ,\cdot \rangle\) be the real dual product of \(H^{1}(\varOmega )\) and \(H^1_0(\varOmega )\). Let \(\mathcal {A}:H^1_0(\varOmega )\rightarrow H^{1}(\varOmega )\) be defined by
The operator \(A:\mathcal {D}(A)\subset L^2(\varOmega )\rightarrow L^2(\varOmega )\) is defined by \(Au:=\mathcal {A}u\) for \(u\in \mathcal {D}(A)\), where \(\mathcal {D}(A):=\{u\in H^1_0(\varOmega )\mid \mathcal {A}u\in L^2(\varOmega )\}\). Let \(J\subset \mathbb {R}\) be a time interval. For any function \(v:J\times \varOmega \rightarrow \mathbb {R}\), we introduce the shortened forms \(v(t):=v(t,\cdot )\) and \(\partial _tv(t):=(\partial _tv)(t,\cdot )\), where \(\partial _t\) denotes the weak derivative of \(t\in J\). For a real Banach space X, the function space C(J; X) is defined by the set of continuous functions as \(J\ni t\mapsto v(t)\in X\). Let \(C^1(J;X)\) be the set of \(C^1(J)\) functions as \(J\ni t\mapsto v(t)\in X\). For a time interval \((a,b)~(0\le a<b<\infty )\), we define the function space:
Assume that \(u_0\in H^1_0(\varOmega )\) and \(1<p<\frac{N+2}{N2}\), where we replaced \(\frac{N+2}{N2}\) by \(\infty\) for \(N=1,2\). Recall that we consider a blowup solution of the Fujitatype equation:
Fujita pioneered in studies of a solution for (2) with \(\varOmega =\mathbb {R}^N\) in 1966 [6]. This seminal work has inspired many researchers to study solutions of variable timeevolution partial differential equations including (2) (see e.g., [13, 26, 27] and references therein). Currently, the existence of solutions of (2) in a bounded time interval \([0,T)~(T<\infty )\) has been proven. For instance, under some assumptions, we can prove the existence of the solution in \(Z_{(0,T)}\) (see e.g., Theorem 3.1 in [1]). For the existence of a blowup solution of (2), the sufficient conditions for the initial function \(u_0\) should be satisfied. It can be achieved using the first eigenvalue of the operator \(\mathcal {A}\) defined in (1) [9], the energy functional (see e.g., [1, 25]), and an unstable stationary solution of (2) (see e.g., [12]), etc. In the numerical analysis, many approximations of the blowup time have been suggested and proved to converge to the exact blowup time (see, e.g., [3, 4, 17]). Computerassisted methods for finding the exact value of the blowup time of the solution for ordinary differential equations have also been established (see, e.g., [14, 15, 23]). However, the same problem for the partial differential equations has not been properly addressed so far.
We define \(L^2(\varOmega )\) blowup solution as a solution of (2) satisfying
for some \(t_{\max }^*>0\). The \(t^*_{\max }\) is also defined as \(L^2(\varOmega )\) blowup time.
We provide an explicit and computable inclusion of the \(L^2(\varOmega )\) blowup time for the solution u of (2) in \(Z_{(0,t_{\max }^*)}\). We propose a timestepping algorithm using numerical verification methods for obtaining a sharp inclusion of the \(L^2(\varOmega )\) blowup time. Our proposed algorithm is initialized at \(t_0=0\), \(i=0\) and includes the following steps:
 Step 1::

Estimate an upper bound \(\overline{t}_{\max }\) of the \(L^2(\varOmega )\) blowup time using the initial value \(u(t_{i})\). Then, note that the true \(L^2(\varOmega )\) blowup time \(t_{\max }^*\) is included in the interval \((t_{i}, \overline{t}_{\max }]\). We proceed to Step 2 even if \(\overline{t}_{\max }\) is not obtained.
 Step 2::

Prove the existence of the solution in the time interval \([t_{i}, t_{i+1}]\) using a computerassisted proof and obtain the solution \(u(t_{i+1})\) at the time \(t_{i+1}\).
 Step 3::

Replace i with \(i+1\) and return to Step 1.
Note that in Step 3, we update the initial value of the Fujitatype equation from \(u(t_i)\) to \(u(t_{i+1})\). Hereinafter, \(i\in \mathbb {N}\cup \{0\}\) in Steps 1, 2, and 3 are called the “step number”.
In Step 2, we prove that the solution exists in time interval \([t_i,t_{i+1}]\) by computerassisted proofs of the existence for solutions of parabolic equations. Since the first derivation by Nakao [18], several methods of the computerassisted proofs of the existence for solutions of parabolic equations have been proposed. These methods have been improved with time and can now verify the existence of solutions for parabolic equations in a longtime interval \(J\subset \mathbb {R}\) [7, 24]. In this paper, we use the method in [7] (see Appendix B for details) in Step 2.
Here, we focus on Step 1. In Step 1, we introduce the \(L^2(\varOmega )\) blowup time estimate using the energy functional \(E:H^1_0(\varOmega )\cap L^{p+1}(\varOmega )\rightarrow \mathbb {R}\) defined below for obtaining the upper bound \(\overline{t}_{\max }\):
Several wellknown theorems provide some sufficient conditions concerned with the value of \(E(u_0)\), where \(u_0\) is an initial function of (2), for proving that the solution u of (2) is the \(L^2(\varOmega )\) blowup solution. However, the exact value of the upper bound of the \(L^2(\varOmega )\) blowup time is not obtained in the wellknown theorems (see e,g., [1]). In Theorem 1, we formulate a theorem for giving an explicit and computable inclusion of the \(L^2(\varOmega )\) blowup time as well as verifying the existence of \(L^2(\varOmega )\) blowup solution by means of numerical verification methods:
Theorem 1
Let \(i\in \mathbb {N}\cup \{0\}\) be the step numbers, \(u(t_i)\) be the solution at the time \(t_i\) in Steps 1, 2, and 3. Also, let \(\varOmega \subset \mathbb {R}^N(N\in \mathbb {N})\) be a bounded Lipschitz domain. Assume that there exists the solution of (2) in \(Z_{(0,t_i)}\cap C([0,t_i];H^1_0(\varOmega ))\). If
holds, then the solution u is the \(L^2(\varOmega )\) blowup solution. Moreover, we obtain the upper bound of \(L^2(\varOmega )\) blowup time as
where \(\varOmega \) is Lebesgue’s measure of \(\varOmega\) and \(\overline{t}_{\max }\) is defined in Step 1.
Note that we can obtain an inclusion of the \(L^{\infty }(\varOmega )\) blowup time if the inequality (3) holds under some assumptions (see Appendix A for details).
In Step 1, we first derive the inclusion of \(\Vert u(t_i)\Vert _{L^2(\varOmega )}\) and \(E(u(t_i))\) using a numerical verification method (see Appendix C for details). Moreover, if the sufficient condition (3) is satisfied, we can compute the upper bound \(\overline{t}_{\max }\) of the \(L^2(\varOmega )\) blowup time using (4).
We can verify the existence of the solution in time interval \([0,t_{i+1}]\) and demonstrate it blowing up for \(t>t_{i+1}\) using iterative procedure. Therefore, our method can verify whether the solution blows up even if we cannot determine that the blowingup occurs from the initial function \(u_0\). We provide the detailed algorithm description in Algorithm 1.
This paper is organized as follows. In Sect. 2, we prove Theorem 1. In Sect. 3, we introduce several numerical examples obtained by the Algorithm 1.
2 Proof of Theorem 1
In the below, we first present some auxiliary lemmas and a corollary, which are used to prove Theorem 1. Corollary 1 is wellknown with a smooth domain \(\varOmega\) (see e.g., [20, Lemma 17.5]). We show Lemma 1 for proving Corollary 1 for the case, when the bounded domain \(\varOmega\) has Lipschitz boundary.
Lemma 1
Let \(0<T<\infty\) and \(\varOmega \subset \mathbb {R}^N(N\in \mathbb {N})\) be a bounded Lipschitz domain. If the solution u of (2) in \(Z_{(0,T)}\) exists,
holds, where \(u_0\) is an initial function of (2).
Proof
Let \(t\in (0,T]\). Multiplying (2a) by \(\partial _t u\in C((0,T);L^2(\varOmega ))\) and integrating \((0,t]\times \varOmega\), we get:
First, for \(u(t)\in \mathcal {D}(A)\), we consider \(\int _{0}^{t}(Au(s),\partial _su(s))_{L^2(\varOmega )}ds\) in (5). Since the inner product is continuous and the Dirichlet Laplace operator A is the selfadjoint on \(L^2(\varOmega )\), we have
Integrating by parts on (0, t), we obtain
Thus, we have
Next, we consider \(\int _{0}^t(u(s)^{p1}u(s),\partial _s u(s))_{L^2(\varOmega )}ds\) in (5). Note that
Integrating by parts on (0, t), we get
Therefore, we end up with
Here, the embedding theorem from \(H^1_0(\varOmega )\) to \(L^{p+1}(\varOmega )\) implies that \(u_0\in L^{p+1}(\varOmega )\) because the exponent p, which is defined by (2), satisfies \(1<p<\frac{N+2}{N2}\) (see e.g., [21, Part 3 of Theorem 4.12]). From (5)–(7) we get
This completes a proof of Lemma 1.
Corollary 1
Let \(0\le s\le t<T<\infty\) and \(\varOmega \subset \mathbb {R}^N(N\in \mathbb {N})\) be a bounded Lipschitz domain. If the solution u of (2) in \(Z_{(0,T)}\) exists,
holds.
Proof
Lemma 1 yields
Therefore, we obtain \(E(u(t))\le E(u(s))\).
We introduce the wellknown Lemma 3, which leads an upper bound of the blowup time in variable topologies for solutions of several differential equations (see e.g., [11, 16, 28]). We provide Lemma 2 for proving Lemma 3:
Lemma 2
(Theorem 3.1 in [2]) Let \(0<x_0\le x_1\). Let \(a,b:[0,\infty )\rightarrow \mathbb {R}\) be positive and continuous functions. Assume that b is nondecreasing and functions u,v: \([0,\infty )\rightarrow \mathbb {\overline{R}}\), where \(\mathbb {\overline{R}}\) is defined by the set of extended real numbers, satisfy
Then, \(v(t)\ge u(t)\) for \(t\ge 0\), and \(T_e^v\le A^{1}(B(\infty ))\), where we define \(T_e^v\ge 0\), \(A:[0,\infty )\rightarrow \mathbb {R}\), and \(B:[x_0,\infty )\rightarrow \mathbb {R}\) as
respectively.
Then we have the following lemma:
Lemma 3
Let \(t_M\ge 0\). Let \(g:\mathbb {R}\rightarrow \mathbb {R}\) be a nondecreasing and continuous function. Suppose that a function \(y\in C([t_M,\infty ))\cap C^1((t_M,\infty ))\) satisfies
Furthermore, we assume that
Then, there exists \(t^*>t_M\) such that
Proof
For \(t\in [t_M,\infty )\), integrating (8) on \([t_M,t]\), we obtain
Replacing \([0,\infty )\), which is the domain of functions a, b, u, v, and A, with \([t_M,\infty )\) and setting \(a(x)=1\), \(b(x)=g(x)\), and \(x_0=x_1=y(t_M)\), respectively in Lemma 2, Lemma 2 implies that the function y satisfies (9).
Now, let us prove Theorem 1.
Proof of Theorem 1
Provided that the solution \(u\in Z_{(0,t_i)}\) of (2) exists, we consider the Fujitatype equation (2) with the initial function \(u_0=u(t_i)\), where \(u(t_i)\) is the solution at the time \(t_i\). Let \(J_\infty =(t_i,\infty )\). First, assuming that the solution \(u\in Z_{J_\infty }\) exists, we show a contradiction. Let \(t\in J_\infty\). Note that the solution \(u\in Z_{J_\infty }\) satisfies
(see, e.g., [5, Theorem 3 in Section 5.9]) and \(\partial _t\) can be treated as the usual derivative (not in the sense of distributions), where \(\frac{d}{dt}\Vert u(t)\Vert _{L^2(\varOmega )}^2\) denotes the derivative of the function \(J_\infty \ni s\mapsto \Vert u(s)\Vert _{L^2(\varOmega )}^2\) at \(s=t\). Multiplying (2a) by u(t) and integrating in \(\varOmega\), we obtain
From (10) and (11) it follows that
Corollary 1 with \(s=t_i\) yields \(E(u(t))\le E(u(t_i))\) for \(t>t_i\). Since \(\varOmega\) is bounded, H\(\ddot{\text{ o }}\)lder’s inequality implies that
Here we accounted for (12). Lemma 3 with \(t_M=t_i\), \(y(t)=\Vert u(t)\Vert _{L^2(\varOmega )}^2\), and \(g(x)=4E(u(t_i))+\frac{2(p1)}{p+1}\varOmega ^{\frac{1p}{2}}x^{\frac{p+1}{2}}\) implies that the \(L^2(\varOmega )\) blowup time \(t_{\max }^*\) satisfies
Note that \(g(x)>0\) for \(x>y(t_i)\) and \(\displaystyle \int _{y(t_i)}^{\infty }g(s)^{1}ds<\infty\) by (3). Therefore, the solution u does not exists in \(Z_{J_\infty }\). This is a contradiction.
Next, we derive an upper bound of the \(L^2(\varOmega )\) blowup time \(t^*_{\max }\). Setting \(s=t^{2}\), we obtain
It follows from (13) that the upper bound \(\overline{t}_{\max }\) of the \(t_{\max }^*\) can be derived from
3 Numerical results
We consider the Fujitatype equation with \(p=2\):
where \(J=(0,T)~(T<\infty )\), a bounded domain \(\varOmega \subset \mathbb {R}\), \(u_0\in \mathcal {D}(A)\), and \(u_0>0\) means that \(u_0(x)>0~a.e.~x\in \varOmega\). We verify that the solution u of (14) blows up in the \(L^2(\varOmega )\) sense and obtain explicit numerical inclusions of the \(L^2(\varOmega )\) blowup time using Algorithm 1. Let \(V_h\) and \(V_k\) be the sets depending on \(h>0\) and \(k>0\) of the piecewise Hermite spline (\(C^1\)class with 5degree) functions in the domain \(\varOmega\) and the piecewise quadratic (\(C^0\)class) functions in J, respectively. We define \(V_{h,k}:=V_h\otimes V_k\) and construct a fulldiscrete approximation \(u_{h,k}\in V_{h,k}\) of (14). The method in [7] verifies the existence of the solution \(u\in Z_J\) of (14) (see Appendix B for details). If the method in [7] succeeds to verify, we can compute explicit error bounds \(\rho _{L^2}\) and \(\rho _{H^1_0}\) satisfying that \(\rho _{L^2}\ge \Vert (uu_{h,k})(t_i)\Vert _{L^2(\varOmega )}\) and \(\rho _{H^1_0}\ge \Vert (uu_{h,k})(t_i)\Vert _{H^1_0(\varOmega )}\) for each step number i. Using \(\rho _{L^2}\) and \(\rho _{H^1_0}\), we can also compute an explicit error bound \(E_{err}\) satisfying that \(E_{err}>E(u(t_i))E(u_{h,k}(t_i))\). We have an upper bound of the energy functional \(E(u(t_i))\) and a lower bound of the \(L^2(\varOmega )\) norm of the forms:
when \(\Vert u_{h,k}(t_i)\Vert _{L^2(\varOmega )}>\rho _{L^2}\) holds. Then, we can check that the condition (3) holds or not, where we write the details of the procedure of checking the condition of (3) in Appendix C.
In this section, we put \(h=1/64\) and \(k=(t_{i+1}t_{i})/128\) for the step numbers i and set the minimal timestep width \(\tau _{\min }=10^{4}\) in Algorithm 1. All computations were carried out on the Dell Precision 7920 Intel Xeon Gold 6134 CPU 3.20GHz with MATLAB (var. R2018a) and Symbolic Math Toolbox. Furthermore, we added INTLAB(var. 10.1), which calculated estimates including all rounding errors [22].
3.1 Numerical example 1
Let \(\varOmega =(0,1)\). We take the initial function \(u_0\) as \(u_0(x)=\frac{192}{5}x(x1)(x^2x1)\) (see Fig. 1).
The approximations \(u_{h,k}\) of (14) with the initial function \(u_0\) were displayed in time intervals [0, 0.0188], [0, 0.2988], [0, 0.3058], and [0.3058, 0.3067] (Fig. 2). The figure shows that the maximum values of the approximations become bigger as the time progresses.
We verified the existence of the solution u, which blows up, and led an explicit inclusion of the \(L^2(\varOmega )\) blowup time using Algorithm 1. For the step numbers i, the detailed estimates of (15), \(\overline{t}_{\max }t_i\), and \((\overline{t}_{\max }t_i)/t_i\), which were obtained by Algorithm 1, are summarized in Table 1, where the “Failed1” means the failure of checking the sufficient condition (3) in Step 1 and the “Failed2” means that we failed to verify the solution u in time \([t_i,t_{i+1}]\) when \(\tau \ge \tau _{\min }\) in Step 2 in Algorithm 1, respectively. When we started from \(t=0\) and the step number \(0\le i\le 8\), it could not be proved that the solution u blows up using the procedure of Step 1 in Algorithm 1; that is, we could not check that the sufficient condition (3) in Theorem 1 holds. The procedure of Step 2 in Algorithm 1 with the computerassisted method given by [7] started to prove that the solution u exists in a neighborhood centered at \(u_{h,k}\) for the step numbers i. When the step number \(i\ge 9\), the procedure of Step 1 in Algorithm 1 proved that the condition (3) in Theorem 1 is satisfied. More specifically, we verified that the solution blows up and obtain the upper bound \(\overline{t}_{\max }\) of the \(L^2(\varOmega )\) blowup time. When \(i=52\), Algorithm 1 were aborted because the timestep width \(\tau\) satisfied \(\tau _{\min }>\tau\) in the procedures of Step 2. We wrote the lower bounds \(t_i\) and the upper bounds \(\overline{t}_{\max }\) obtained by Algorithm 1 in Table 2 and plotted them in Fig. 3. It demonstrated that the absolute errors \(\overline{t}_{\max }t_i\) became smaller for \(i\le 46\) compared with the corresponding values in the previous steps. However, for \(i\ge 47\), because of the fulldiscrete approximate solutions \(u_{h,k}\) with low accuracy, the absolute errors have been bigger compared with the corresponding values in the previous steps (Table 1). We have that the sharpest inclusion of the \(L^2(\varOmega )\) blowup time is in the interval (0.3068, 0.317713] from results corresponding with \(i=52\) and \(i=46\) in Table 2, respectively. The inclusion result is mathematically interesting because we cannot show that the solution u in this numerical example blows up in the \(L^2(\varOmega )\) sense using the existing results (see e.g., [20, Theorem 17.6]).
3.2 Numerical example 2
Let \(\varOmega =(0,1)\). We take the initial function \(u_0\) as
(see Fig. 4)
The fulldiscrete approximations \(u_{h,k}\) of (14) with the initial function \(u_0\) in time intervals [0, 0.0188], [0, 0.2188], [0, 0.2278], and [0.2278, 0.2284] are shown in Fig. 5.
The maximal value of approximation \(u_{h,k}\) became smaller in the time interval [0, 0.0188]. However, the value grew for the time \(t>0.0188\). We proved that the solution u exists and blows up using Algorithm 1. Moreover, we led an explicit inclusion of the \(L^2(\varOmega )\) blowup time by Algorithm 1. For the step numbers i, estimates for (15), \(\overline{t}_{\max }t_i\), and \((\overline{t}_{\max }t_i)/t_i\), which were provided by Algorithm 1, are written in Table 3, where the “Failed1” and “Failed2” are defined in Sect. 3.1. The procedure of Step 1 in Algorithm 1 could not check that the condition (3) in Theorem 1 holds at the time \(t=0\). The procedure of Step 2 in Algorithm 1 with the computerassisted method by [7] started to verify the existence of the solution u in a neighborhood centered at \(u_{h,k}\) for the step numbers i. When the step number \(i\ge 8\), we could prove that the solution u blows up and compute the upper bound \(\overline{t}_{\max }\) of the \(L^2(\varOmega )\) blowup time using the procedure of Step 1 in Algorithm 1. However, when \(i=43\), Algorithm 1 was aborted because the method by [7] could not prove the existence of the solution u for the time \(t>t_i\) even if we set \(\tau =\tau _{\min }\) in the procedures of Step 2. For the step numbers i, the lower bounds \(t_i\) and the upper bounds \(\overline{t}_{\max }\) obtained by the method in Algorithm 1 were shown in Table 4 and plotted them in Fig. 6, respectively. It yields that the absolute errors \(\overline{t}_{\max }t_i\) become smaller for \(i\le 36\) and bigger for \(i\ge 37\) compared with the corresponding values in the previous steps. For \(i\ge 37\), the approximate solutions \(u_{h,k}\) with low accuracy prevent from deriving the sharp inclusion of the \(L^2(\varOmega )\) blowup time (Table 3). The sharpest inclusion of the \(L^2(\varOmega )\) blowup time is in the interval (0.2285, 0.257074] from the results with \(i=43\) and \(i=36\) in Table 4, respectively.
4 Conclusion
We have proposed a timestepping algorithm using numerical verification methods for obtaining explicit inclusion of the \(L^2(\varOmega )\) blowup time of a solution to the Fujitatype equation. The algorithm consists of three iteration steps. Step 1 provides a criterion whether the solution u blows up at \(t>t_i\) as well as a rigorous upper bound of \(\overline{t}_{\max }\). Moreover, we have formulated Theorem 1 for constructing the detailed procedure of Step 1. In Step 2, we have verified the existence of the solution in each time interval \([t_i,t_{i+1}]\) using a computerassisted existential proof of solutions for parabolic equations. In Step 3, we have updated the step number i to \(i+1\) and attempted to verify the \(L^2(\varOmega )\) blowup solution for time \(t>t_{i+1}\) by Step 1. Our method not only verifies that the solution blows up but also derives the explicit inclusion of the \(L^2(\varOmega )\) blowup time by iterating the procedures of Steps 1, 2, and 3. Our method remains valid even when we cannot determine whether the solution blows up in finite time from the initial function \(u_0\). We believe that our method will find applications in other problems involving various partial differential equations.
References
Ball, J.M.: Remarks on blowup and nonexistence theorems for nonlinear evolution equations. Q. J. Math. 28(4), 473–486 (1977). (12)
CeballosLira, M.J., MacíasDíaz, J.E., Villa, J.: A generalization of Osgood’s test and a comparison criterion for integral equations with noise. Electron. J. Diff. Equ. 2011(5), 1–8 (2011)
Hirota, C., Ozawa, K.: Numerical method of estimating the blowup time and rate of the solution of ordinary differential equationsan application to the blowup problems of partial differential equations. J. Comput. Appl. Math. 193(2), 614–637 (2006)
Cho, C.H.: A numerical algorithm for blowup problems revisited. Numer. Algorithms 75, 675–697 (2017)
Evans, L.C.: Partial Differential Equations, 2nd edn. American Mathematical Society, Providence (2010)
Fujita, H.: On the blowing up of solutions of the Cauchy problem for \(u_t\Delta u=u^{1+\alpha }\). J. Fac. Sci. Univ. Tokyo. Sect. 1 13(2), 109–124 (1966)
Hashimoto, K., Kinoshita, T., Nakao, M.T.: Numerical verification of solutions for nonlinear parabolic problems. Numer. Funct. Anal. Optim. 41(12), 1495–1514 (2020)
Jerison, D., Kenig, C.E.: The Inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal. 130(1), 161–219 (1995)
Kaplan, S.: On the growth of solutions of quasilinear parabolic equations. Commun. Pure Appl. Math. 16(3), 305–330 (1963)
Kinoshita, T., Kimura, T., Nakao, M.T.: On the a posteriori estimates for inverse operators of linear parabolic equations with applications to the numerical enclosure of solutions for nonlinear problems. Numer. Math. 126, 679–701 (2014)
Kou, W., Ding, J.: Global existence and blowup analysis for parabolic equations with nonlocal source and nonlinear boundary conditions. Boundary Value Probl 2020, 37 (2020). https://doi.org/10.1186/s13661020013405
Lions, P.L.: Asymptotic behavior of some nonlinear heat equations. Phys. D 5(2), 293–306 (1982)
Liu, X., Zhang, T.: \(H^2\) blowup result for a Schrödinger equation with nonlinear Sorce term. Am. Inst. Math. Sci. 28, 777–794 (2020)
Matsue, K., Takayasu, A.: Numerical validation of blowup solutions with quasihomogeneous compactifications. Numer. Math. 145, 605–654 (2020)
Matsue, K., Takayasu, A.: Rigorous numerics of blowup solutions for odes with exponential nonlinearity. J. Comput. Appl. Math. 374, 112607 (2020). https://doi.org/10.1016/j.cam.2019.112607
Na, Y., Zhou, M., Zhou, X., Gai, G.: Blowup theorems of Fujita type for a semilinear parabolic equation with a gradient term. Adv. Differ. Equ. 2018, 128 (2018). https://doi.org/10.1186/s1366201815822
Nakagawa, T.: Blowing up of a finite difference solution to \(u_t = u_{xx} + u^2\). Appl. Math. Optim. 2, 337–350 (1975)
Nakao, M.T.: Solving nonlinear parabolic problems with result verification. Part I: onespace dimensional case. J. Comput. Appl. Math. 38(1), 323–334 (1991)
Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012)
Quittner, P., Souplet, P.: Superlinear Parabolic Problems: Blowup, Global Existence and Steady States, 2nd edn. Birkhäuser, Boston (2019)
Adams, R.A.: Sobolev Spaces, 2nd edn. Academic Press, New York (1975)
Rump, S.M.: INTLAB—INTerval LABoratory. In: Csendes, T. (ed.) Developments in Reliable Computing, pp 77–104. Kluwer Academic Publishers, Dordrecht (1999)
Takayasu, A., Matsue, K., Sasaki, T., Tanaka, K., Mizuguchi, M., Oishi, S.: Numerical validation of blowup solutions of ordinary differential equations. J. Comput. Appl. Math. 314, 10–29 (2017)
Takayasu, A., Mizuguchi, M., Kubo, T., Oishi, S.: Accurate method of verified computing for solutions of semilinear heat equations. Reliab. Comput. 25, 74–99 (2017)
Tsusumi, M.: On solutions of semilinear differential equations in Hilbert space. Math. Jpn. 17, 173–193 (1972)
Wang, H., He, Y.: On blowup of solutions for a semilinear parabolic equation involving variable source and positive initial energy. Appl. Math. Lett. 26(10), 1008–1012 (2013)
Zhang, P.: Global Fujita–Kato solution of 3D inhomogeneous incompressible Navier–Stokes system. Adv. Math. (N. Y.) 363, 107007 (2020). https://doi.org/10.1016/j.aim.2020.107007
Zhou, Y.C., Yang, Z.W., Zhang, H.Y., Wang, Y.: Theoretical analysis for blowup behaviors of differential equations with piecewise constant arguments. Appl. Math. Comput. 274, 353–361 (2016)
Acknowledgements
We appreciate editors in this journal and anonymous reviewers’ useful comments for improving quality of this paper. This work was supported by CREST, JST Grant no. JPMJCR14D4, JSPS KAKENHI no. 18K13462, JSPS KAKENHI no. 18K03434, JSPS KAKENHI no. 18K03440, and JSPS KAKENHI no. 21H00998.
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Appendices
Appendix A: An inclusion of \(L^{\infty }(\varOmega )\) blowup time in Theorem 1
Theorem 1 provides not only \(L^{2}(\varOmega )\) blowup time but also \(L^{\infty }(\varOmega )\) blowup time. We use notations \(t_i\) and \(\overline{t}_{\max }\) defined in Theorem 1. Let the \(L^{\infty }(\varOmega )\) blowup time \(t_{\infty }>0\) defined by
where \(u\in Z_{(0,t_i)}\cap C([0,t_i];H^1_0(\varOmega ))\) is the solution of (2) and \(\Vert u(t)\Vert _{L^{\infty }(\varOmega )}:=\displaystyle \mathop {\hbox {ess}\,\hbox {sup}}\limits _{x\in \varOmega }u(t,x)\). Assume that \(D(A)\subset L^{\infty }(\varOmega )\) holds. For example, suppose that \(N\le 2\) (see e.g., [8] and Theorem 8.2 in [19]), where \(N\in \mathbb {N}\) is the dimension of the domain \(\varOmega\).
We consider an inclusion of \(t_\infty\). Since \(u\in Z_{(0,t_{i})}\subset C((0,t_{i});\mathcal {D}(A))\subset C((0,t_{i});L^{\infty }(\varOmega ))\), we have \(t_i\le t_{\infty }\). If \(t_{\infty }>t^*_{\max }\), where \(t^*_{\max }\) is the \(L^2(\varOmega )\) blowup time, we have a contradiction because H\(\ddot{\text{ o }}\)lder’s inequality implies that
Therefore, the estimate \(t_{\infty }\le t^*_{\max }\) holds and we obtain that the \(L^{\infty }(\varOmega )\) blowup time \(t_{\infty }\) is included in \([t_i,\overline{t}_{\max }]\) by \(t^*_{\max }\le \overline{t}_{\max }\).
Appendix B: The verification method in [7]
We briefly present a numerical verification method of solutions for the nonlinear problem (14) with \(g(u):=uu\). Note that \(g(u)=u^2\) since \(u(t)\ge 0\) for all \(t>0\) on \(\varOmega\) by the assumption \(u_0>0\) in (14). Dividing interval J into l subintervals, we use a time evolving algorithm in the below.
In order to get an approximate solution for the problem (14) on \(\varOmega \times J\), we use a finite element subspace \(V_{h,k}(\varOmega ,J):=V_h(\varOmega )\otimes V_k(J)\), where we suppose that \(V_k(J)\subset H^1(J)\) and \(V_h(\varOmega )\subset H^2(\varOmega )\), respectively. Let \(u_{h,k}^{(i)}\in V_{h,k}(\varOmega ,J_i)\) be an approximate solution of the problem (14) on \(\varOmega \times J_i\), where \(J_i=(t_{i1},t_{i})\) is a subinterval of J with \(t_0 =0\), and \(T_i\equiv J_i=t_it_{i1}\) for \(i=1,\cdots ,l\).
Letting \(\bar{u}:=uu_{h,k}^{(i)}\), the problem (14) is equivalent to the following residual equation:
where \(\epsilon _{i}:=u(t_{i1})u_{h,k}^{(i1)}(t_{i1})\) with \(\epsilon _{1}=u_{0}u_{h,k}^{(1)}(t_0)\) and \(\delta _{i}:=g(u_{h,k}^{(i)})\partial _{t}u_{h,k}^{(i)}+\nu \varDelta u_{h,k}^{(i)}\) is a residual function for \(i=1,\ldots ,l\).
We define the operators \(\mathcal {L}_{i}: Z_{J_i} \rightarrow L^2\bigl (J_i;L^2(\varOmega )\bigr )\) by
where \(g^{\prime }[u_{h,k}^{(i)}]=2u_{h,k}^{(i)}\) denote the Fréchet derivative of g at \(u_{h,k}^{(i)}\). Using the operators \(\mathcal {L}_{i}\), a solution \(\bar{u}\) of the problem (16) can be decomposed as \(\bar{u}=v+w\) by using solutions v and w of
and
respectively, where \(g_i(w)\equiv g(v+w+u_{h,k}^{(i)})g(u_{h,k}^{(i)})g^{\prime }[u_{h,k}^{(i)}](v+w)+\delta _{i}\) for \(i=1,\cdots ,l\). Since the solution v of (17) can be determined independently of w in (18), if the solution v of the linear problem (17) is numerically verified, then the problem (16) can be reduced to find a solution w to the nonlinear problem (18). First, we present a numerical verification method of solutions for nonlinear problems (18). Note that the problem (18) is rewritten as the following fixedpoint equation of the map \(\mathcal {L}_{i}^{1}\):
Here, the map \(\mathcal {L}_{i}^{1}: L^2\bigl (J_i;L^2(\varOmega )\bigr ) \rightarrow Z_{J_i}\) is considered as the solution operator for the linear parabolic problem with homogeneous initial condition.
For any positive constants \(\alpha _i\) and \(\beta _i\), we define the candidate set \(W_{\alpha _i,\beta _i}\) of solutions in (18) as
where \(H_0^1(\varOmega ):=\{u \in H^1(\varOmega ),\ u = 0 \text { on } \partial \varOmega \}\), \(V^1(J_i):=\{u \in H^1(J_i),\ u(t_{i1}) = 0\}\) and \(V(\varOmega ,J_i)\equiv V^1\bigl (J_i;L^2(\varOmega )\bigr ) \cap L^2\bigl (J_i;H_0^1(\varOmega )\bigr )\).
The map \(\mathcal {I}\mathcal {L}_{i}^{1}:L^2(J_i;L^2(\varOmega ))\rightarrow L^2(J_i;H^1_0(\varOmega ))\) is continuous and compact (see e.g., Section 2 in [10]), where \(\mathcal {I}\) is the inclusion map from \(Z_{J_i}\) to \(L^2(J_i;H^1_0(\varOmega ))\). From the Schauder fixedpoint theorem, if the set \(W_{\alpha _i,\beta _i}(\varOmega ,J_i)\) satisfies
then a fixedpoint of (19) exists in the set \(\overline{W_{\alpha _i,\beta _i}(\varOmega ,J_i)}\), where \(\overline{W_{\alpha _i,\beta _i}(\varOmega ,J_i)}\) stands for the closure of the set \(W_{\alpha _i,\beta _i}(\varOmega ,J_i)\) in \(L^2(J_i;H^1_0(\varOmega ))\). In order to estimate \(\mathcal {I}\mathcal {L}_{i}^{1}g_i(w)\) for \(w \in W_{\alpha _i,\beta _i}(\varOmega ,J_i)\), setting \(\tilde{w} : = \mathcal {I}\mathcal {L}_{i}^{1}g_i(w)\), we can obtain
where constants \({{\mathcal {M}}}_{1}^{(i)}\), \({{\mathcal {M}}}_{t_i}^{(i)}\) and \(C_{\varDelta }^{(i)}\) are numerically determined which presented by Lemma 4 and Theorem 5 in [7]. Moreover, by Lemma 6 in [7], the solution v of (17) is estimated as follows:
where \(\rho (T_i): =\exp (\pi ^2T_i)\) and \(\rho _{\varOmega }(T_i):=\sqrt{\frac{1}{2\pi ^2}(1\rho (2T_i))}\), if \(\varOmega =(0,1)\).
Therefore, defining the function \(G(\alpha _i,\beta _i)\) of \(\alpha _i\) and \(\beta _i\) satisfying
the sufficient condition of (20) is given as follows:
Next, we describe several remarks on the verification step from the interval \(J_i\) to \(J_{i+1}\). Let \(\alpha ^*_{i}\) and \(\beta ^*_{i}\) be two positive numbers satisfying the condition (23). Then there exists a solution \(w_*\in W_{\alpha ^*_{i},\beta ^*_{i}}(\varOmega ,J_i)\) of the problem (18) and the following estimates hold
When denoting \(v_*\) as a solution of the problem (17), the solution \(u_*\) of the nonlinear problem (14) on \(\varOmega \times J_i\) can be written by \(u_*=u_{h,k}^{(i)}+v_*+w_*\). Note that the initial condition of the next timestep problem on \(\varOmega \times J_{i+1}\) is given by \(u_*(t_i)=u_{h,k}^{(i)}(t_i)+v_*(t_i)+w_*(t_i)\). Since we take the approximate solution \(u_{h,k}^{(i+1)}\in V_{h,k}(\varOmega ,J_{i+1})\) satisfying \(u_{h,k}^{(i+1)}(t_i)=u_{h,k}^{(i)}(t_i)\), an initial function \(\epsilon _{i+1}\) of the problem (16) on \(\varOmega \times J_{i+1}\) is given by
Therefore, we can obtain the following estimations:
Appendix C: The procedure for checking the condition (3)
For checking the condition (3) with \(p=2\), we show the procedure for the upper bound of \(E(u(t_i))\) and the lower bound \(\Vert u(t_i)\Vert _{L^3(\varOmega )}^3\) in the below.
Let \(u\in Z_{j}\) and \(u_{h,k}\in V_{h,k}(\varOmega ,J_{i})\) be an exact and an approximate solutions of (14) on \(\varOmega \times J_{i}\), respectively. Then we have \(u(t_i)=u_{h,k}(t_i)+\epsilon _{i+1}\), where \(\epsilon _{i+1}\) is defined in (24). Since \(u(t)\ge 0\) for all \(t\in J_i\) on \(\varOmega\), it follows that
Note that \(\Vert u(t_i)\Vert _{H^1_0(\varOmega )}\le \Vert u_{h,k}(t_i)\Vert _{H^1_0(\varOmega )}+\rho _{H^1_0}\) by (25). Thus, we can obtain the following estimation.
where \(E_{err_{H^1_0}}:=\rho _{H^1_0}\left( \rho _{H^1_0}+2\Vert u_{h,k}(t_i)\Vert _{H^1_0(\varOmega )}\right)\). Moreover, since \(u(t_i)=u_{h,k}(t_i)+\epsilon _{i+1}\), we have
where \(E_{err_{L^3}}:=\rho _{L^2}\left( C_{L^\infty }\rho _{H^1_0}\rho _{L^2}+3\left( \Vert u_{h,k}(t_i)^2\Vert _{L^2(\varOmega )}+C_{L^\infty }\rho _{H^1_0}\Vert u_{h,k}(t_i)\Vert _{L^2(\varOmega )}\right) \right)\), \(C_{L^\infty }=\frac{1}{2}\) is a constant satisfying \(\Vert \phi \Vert _{L^{\infty }(\varOmega )}\le C_{L^\infty }\Vert \phi \Vert _{H^1_0(\varOmega )}\) for all \(\phi \in H^1_0(\varOmega )\), and we use the fact \(\varOmega =(0,1)\). Thus, from (26) and (27), we can obtain
On the other hand, it follows that
Hence, if \(\varOmega =(0,1)\) then the sufficient condition of (3) is given as follows:
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Mizuguchi, M., Sekine, K., Hashimoto, K. et al. Rigorous numerical inclusion of the blowup time for the Fujitatype equation. Japan J. Indust. Appl. Math. 40, 665–689 (2023). https://doi.org/10.1007/s13160022005458
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DOI: https://doi.org/10.1007/s13160022005458