Well-posedness of Heat Equations with Nonlinearities of Arbitrarily Rapid GrowthThanks: ^†Corresponding author

Recall (see Definition 2.4) that the Orlicz space $L^{\Phi}=L^{\Phi}(\mathbb{R}^{n})$ consists of Lebesgue measurable functions ${\varphi}$ for which $\Phi(k|{\varphi}|)$ is integrable over $\mathbb{R}^{n}$ for some $k>0$, while the Morse-Transue subspace $M^{\Phi}=M^{\Phi}(\mathbb{R}^{n})$ requires integrability of $\Phi(k|{\varphi}|)$ for all $k>0$, where $\Phi{\colon\!}[0,\infty]\!\to\![0,\infty]$ is some nondecreasing, convex Young’s function with $\Phi(0)=0$ (see Definition 2.1). For ${\varphi}\in L^{\Phi}$, the behaviour of $\Phi$ near zero imposes restrictions on the decay of ${\varphi}$ at spatial infinity while the growth of $\Phi$ at infinity imposes restrictions on the strength of any local singularities of ${\varphi}$. Orlicz space is the natural choice of Banach space in which to generalise the $L^{q}$-theory of [7, 57, 58], with $L^{q}$ corresponding to the special case $\Phi(u)\!=\!u^{q}/q$.

For expositional convenience we refer to any polynomially dominated nonlinearity satisfying $|f(u)|\leq C|u|^{p}$ for some $p>1$ and $u$ large enough, as P-type and any $f$ satisfying $|u|^{p}/|f(u)|\to 0$ as $|u|\to\infty$ for all $p>1$, as FP-type (faster than polynomial). Currently the only general theory of local well-posedness for (NHE) in Banach spaces for a broad class of nonlinearities is that of P-type [7, 31, 32, 49, 57, 58] and FP-type with growth dominated by $\exp(u^{p})$ [25, 37]. The corresponding problem for global existence with small initial data has received more attention [18, 38, 37].

In any general framework seeking a local solution semiflow ${\mathscr{U}}$, one would like $S(t)$ to be a $C_{0}$-semigroup on the underlying space $X$ so that a continuous solution trajectory $u\in C([0,T_{{\varphi}}),X)$ might be obtained for every ${\varphi}\in X$. However, for Orlicz spaces $L^{\Phi}$ this is generally not the case since the set of smooth compactly supported functions $C_{0}^{\infty}(\mathbb{R}^{n})$ are not always dense within them (see e.g., [25, Section 2] where $\Phi(u)=\exp(u^{2})-1$). Hence one cannot hope to obtain a general well-posedness theory requiring $u\in C([0,T),X)$ and without a priori growth restrictions on $f$, by taking $X=L^{\Phi}$. Under more stringent growth restrictions on $f$ such as those of P-type (e.g., $f(u)=|u|^{p-1}u$), it is well-known ([7, 49, 57, 58]) that one can obtain a well-posedness theory in Lebesgue space $L^{\Phi}=L^{q}$ due to the fact that $C_{0}^{\infty}$ is dense in $L^{q}$ for $1\leq q<\infty$; but without such restrictions a general well-posedness theory in $L^{\Phi}$ is not possible.

The obvious strategy therefore is to work in $X=L_{0}^{\Phi}$ (as in e.g., [26, 37]), where $L_{0}^{\Phi}$ denotes the closure of $C_{0}^{\infty}$ in $L^{\Phi}$ (Definition 2.4(b)). Since we also want to allow singular (unbounded) initial data we do not want to restrict $L^{\Phi}$ to being a subset of $L^{\infty}$ and therefore (see Remark 2.1(a)) we want $\Phi$ to be finite-valued. In such case (see Lemma 4.2(i)) $L_{0}^{\Phi}\subseteq M^{\Phi}$ and so working with initial data in $L_{0}^{\Phi}$ necessarily constrains those data to lie in $M^{\Phi}$. In fact, under the mild restriction that $\Phi$ be an $N$-function (a finite, positive Young’s function satisfying $\Phi(u)/u\to 0$ as $u\to 0$ and $\Phi(u)/u\to\infty$ as $u\to\infty$ - see Definition 2.2 and [1, Section 8]) there is a convenient characterisation of $L_{0}^{\Phi}$ available, namely $L_{0}^{\Phi}=M^{\Phi}$ (see Lemma 4.2(ii)). Since this restriction on $\Phi$ is so mild and the characterisation of $L_{0}^{\Phi}$ via $M^{\Phi}$ so convenient, we work mainly with initial data in $M^{\Phi}$ throughout this paper.

We now describe our first key results concerning the heat semigroup. Their precise statements, together with subsidiary results, are given in Section 3.1. By ‘compatible Young’s functions’ we mean $\Phi$ and $\Psi$ satisfying (3.1).

• Theorem A and Corollary A (outline). For compatible Young’s functions $\Phi$ and $\Psi$, the heat semigroup $S(t){\colon\!}L^{\Phi}\!\to\!L^{\Psi}$ satisfies

  $$\left\|S(t){\varphi}\right\|_{\Psi}\leq{\mathscr{C}}\left\|{\varphi}\right\|_{\Phi}\frac{{{\Phi^{-1}}(t^{-\frac{n}{2}})}}{{\Psi^{-1}}(t^{-\frac{n}{2}})},\qquad t>0.$$

  In particular, if $\Phi$ is an $N$-function then $S(t)$ is a $C_{0}$-semigroup on $M^{\Phi}$ and for all $t>0$ we have the $M^{\Phi}$-$L^{\infty}$ smoothing estimate

  $$\left\|S(t){\varphi}\right\|_{\infty}\leq{\mathscr{C}}\left\|{\varphi}\right\|_{\Phi}{\Phi^{-1}}\left(t^{-\frac{n}{2}}\right),\qquad{\varphi}\in M^{\Phi}.$$

Knowing explicitly the dependence on $\Phi$ of the $L^{\infty}$-smoothing estimate enables us to determine a sufficient condition on $\Phi$ and $f$ for the application of the contraction mapping theorem to the integral formulation (1.1), yielding the existence of a classical solution. This estimate is also key to obtaining unconditional uniqueness and continuous dependence of classical solutions, akin to those of the $L^{q}$ case for polynomially dominated nonlinearities in [7], which we prove without further recourse to the contraction mapping theorem. Additionally, for monotonically increasing $f$ we establish a comparison principle for the solution semiflow which is of interest in its own right in the context of monotone dynamical systems theory. With little extra effort, we then establish a sufficient condition for global-in-time well-posedness of (NHE) for small initial data in $M^{\Phi}$. This latter topic has been of some interest in the literature concerned with exponential nonlinearities, especially in its relation to nonlinear Schrödinger equations and the Trudinger-Moser embedding [23, 38, 37, 45, 55]. This appears to be the source of interest in nonlinearities and spaces with growth like $\exp(u^{2})$.

We next outline our first main well-posedness results, their precise statements are given in Section 3.2.

• Theorem B and Corollary B (outline). Let $\Phi$ be an $N$-function. If there exists ${\lambda}>0$ such that

  (1.2)

  $$\int_{1}^{\infty}x^{-p^{*}}\ell\left({\lambda}{\Phi^{-1}}\left(x\right)\right)\,{\rm d}x<\infty,$$

  where $p^{*}\!:=\!1+2/n$ is the Fujita exponent and $\ell$ is the Lipschitz modulus of $f$, then for every ${\varphi}\in M^{\Phi}$ (NHE) possesses a unique classical solution $u$ depending continuously on ${\varphi}$. For $f$ increasing the solution is order-preserving in ${\varphi}$, so that a comparison principle holds. Subject to the stronger condition

  (1.3)

  $$\int_{0}^{\infty}x^{-p^{*}}\ell\left({\lambda}{\Phi^{-1}}\left(x\right)\right)\,{\rm d}x<\frac{n}{4},$$

  for every ${\varphi}$ in a sufficiently small ball in $M^{\Phi}$ there exists a unique global-in-time classical solution $u$ of (NHE) satisfying $u(t)\to 0$ in $L^{\infty}$ as $t\to\infty$.

A common feature of semilinear heat equations is that local existence of solutions is governed only by the growth of the nonlinearity $f$ at infinity [12, 25, 31, 58], while global existence also imposes restrictions on $f$ near zero [23, 33, 37, 38, 59]. We see this reflected in the integral conditions (1.2) and (1.3) respectively, which arise naturally when estimating the integral term in (1.1) via smoothing estimates for the heat semigroup (such as those in Theorem A and Corollary A) in order to obtain a contraction mapping.

The power of Theorem B lies in its flexibility and the fact that no particular structural properties or limitation on growth rates of $f$ or $\Phi$ are imposed a priori. Instead, (1.2) represents a kind of ‘compatibility condition for existence’ between the nonlinearity $f$ and the underlying space $M^{\Phi}$ specified by $\Phi$. It generalises the one obtained for P-type nonlinearities in [31, Theorem 4.4] in the special case $L^{\Phi}=L^{1}$. It also bears a formal resemblance to [58, Eqn. (2.2)] (indeed there is a clear lineage to that work) but there are two crucial qualitative distinctions. Firstly, the definition of $\ell$ in [58] is for the Nemytskii operator associated with $f$, acting on some unspecified function space for which one has no explicit representation. Secondly, the smoothing estimate in [58] is assumed to be of power law type and therefore, as illuminated by Theorem A(b), effectively limits $\Phi$ to one of power law type (i.e., Lebesgue spaces) and consequently $f$ to be of P-type.

We use (1.2) in Theorem B to establish, for a given $f$, how one should choose $\Phi$ in order to obtain local well-posedness of (NHE). To achieve this we impose some structural conditions on $f$ such as convexity for $u\geq 0$ and regular or rapid variation of $\log f$ at infinity. We also assume that $f$ satisfies

(see Section 1.2 for additional context regarding (1.4)). For the sake of brevity at this point we refer the reader to Section 3.3 for the full details of these conditions in (C), $\boldsymbol{(\Lambda)}$ and (G). Below we aim to summarise the essence of these assumptions and their consequences for (NHE).

Consider for the moment the class of nonlinearities of the form

where $\rho>0$ is a constant and $\mu$ is a slowly varying function (see [6, 56] and the discussion around (2.5)). In the terminology of regularly varying functions, if $f$ satisfies (1.5) then $\log f$ is said to be regularly varying of index $\rho$. For $\rho\in(0,1)$ the growth of $f$ is commonly referred to as ‘sub-exponential’, $\rho=1$ as exponential [48] and $\rho>1$ as ‘super-exponential’ (the nonlinearities in [23, 25, 38, 37, 45, 55] are all of this latter type with $\mu\equiv 1$).

In order to treat nonlinearities of arbitrarily large growth rate, we wish to formally permit $\rho=\infty$. We do this rigorously by allowing $\log f$ to be a rapidly varying function (see [56] and around (2.6)), for which there is no upper bound on the growth rate of $f$.

With this in mind we outline our third main theorem. We write $g\lesssim h$ whenever $g(x)\leq h(Kx)$ for large $x$ and some constant $K>0$ and $g\gtrsim h$ whenever $h\lesssim g$. Likewise ‘$g\lesssim h$ as $x\to 0$’ (resp. on $[0,\infty)$), etc. if the relations hold for small $x$ (resp. all $x\geq 0$).

• Theorem C and Corollary C (outline). Let $\Phi$ be an $N$-function and $f$ satisfy (C) and (G).

  • (a)

    If $f$ satisfies $\boldsymbol{(\Lambda)}$ and $\Phi\gtrsim f$ at infinity then (NHE) is locally well-posed in $M^{\Phi}$. If in addition $\Phi\gtrsim f^{r}$ near zero, for some $r\geq 1$ and $r>n/2$, then uniqueness of the classical solution holds in the class $C([0,T],M^{\Phi})$ of mild solutions. Conversely, if $\Phi\lesssim f$ then there exist nonnegative initial data in $L^{\Phi}$ for which no nonnegative solution of (NHE) exists.

  • (b)

    Suppose $f$ satisfies $\boldsymbol{(\Lambda_{0})}$ and that $f^{\prime}$ is regularly varying at zero of index $m-1$ for some $m>p^{*}$. If $\Phi\in\mathcal{N}$ satisfies $\Phi\gtrsim f^{q}$ near zero for some $0<q<n(m-1)/2$ and $\Phi\gtrsim f$ at infinity, then (NHE) is locally well-posed in $M^{\Phi}$ and globally well-posed for small initial data in $M^{\Phi}$.

Theorem C(a) resolves two parts of the central problem. Firstly, for a given $f$ growing at least sub-exponentially but otherwise without dominating growth restrictions, we obtain $\Phi\gtrsim f$ at infinity as an explicit sufficient condition for local well-posedness. Secondly, the nonexistence result shows that this condition is sharp (critical) in the sense that even with $\Phi\approx f$ at infinity (where $g\approx h$ if and only if $g\lesssim h$ and $h\lesssim g$), nonexistence of positive solutions can occur for ${\varphi}$ in the superspace $L^{\Phi}$. The latter is obtained by verifying that a specific initial condition $\varphi$ (given by (6.14)) constructed explicitly in [11] in terms of the function $F$, belongs to $L^{\Phi}$. In this sense any $\Phi$ satisfying $\Phi\approx f$ at infinity is a critical Young’s function. In our terminology above this means that $X_{c}=M^{f}$. This type of criticality has been observed previously in [25, 26] but only in the special case where $f$ has super-exponential growth with $\rho=2$. The significance of Theorem C is that it holds without any upper growth restriction on $f$, i.e., $\rho=\infty$ is permitted.

Theorem C(b) addresses the global aspects of the central problem by providing an additional sufficient order relation near zero for the global well-posedness of (NHE) with small initial data in the critical space $X_{c}=M^{f}$ (note that by Lemma 6.1(iii) $f^{k}\approx f$ at infinity).

In Theorem 7.1 we consider odd nonlinearities of the form $f(u)=u^{m}J(u)$ (for $u\geq 0$) satisfying the conditions of Theorem C and Corollary C. Thus $u^{m}$ captures the behaviour of $f$ near zero while $J$ captures the behaviour of $f$ at infinity. Of particular significance this family contains nonlinearities of arbitrarily rapid growth. Specifically, for any positive increasing convex function $r$ there is a corresponding $J(u)\geq\exp(r(u))$ for which our well-posedness results apply (see Lemma 7.2). We also present additional examples of both FP-type and P-type and compare with known results.

For several decades the most influential work on the well-posedness question for (NHE) has been the $L^{q}$-theory of [57, 58] who considered nonlinearities with growth dominated at large values by a power law function $|u|^{p}$ and $X\!=\!L^{q}$. There, for $p>p^{*}$, the condition $q\geq q_{c}:=n(p-1)/2$ was shown to be sufficient for local well-posedness in $L^{q}$ (in a restricted sense regarding uniqueness, later removed in [7]). Moreover, for $f$ minorised by $u^{p}$, the condition $q\geq q_{c}$ was also shown to be necessary for existence of positive solutions. In this sense the condition on $q$ was sharp, and $L^{q_{c}}$ considered to be a ‘critical space’ (though this terminology originated in a different context related to norm-preservation in $L^{q}$ under a scaling invariance of (NHE) when $f(u)=|u|^{p-1}u$). Viewing $L^{q}$ as an Orlicz space $L^{\Phi}$ with Young’s function $\Phi(u)=u^{q}/q$, one may equivalently consider the function $\Phi_{c}(u)=u^{q_{c}}/{q_{c}}$ to be a ‘critical Young’s function’.

An extensive literature has since developed on the well-posedness of (NHE) in Lebesgue spaces for nonlinearities of P-type [7, 31, 32, 53, 57, 58, 59]. For an overview of its development and the state of the art, [49] is a comprehensive source. This problem has also been studied in Sobolev spaces [22, 52], Besov spaces [40] and a variety of other spaces [19, 24, 29]. However, for nonlinearities of FP-type and framed in Banach spaces a similar corpus does not exist, with only special cases (i.e., exponential $f$) considered.

Coming some thirty-five years after [58], the first step-change in local well-posedness theory for (NHE) dealing successfully with nonlinearities of FP-type was the work of [25]. There, important advances were made for space dimension two $(n\!=\!2)$ with exponentially dominated nonlinearities of the form

and $X=L^{\Phi}=\exp L^{2}(\mathbb{R}^{2})$, $\Phi(u)=\exp(u^{2})-1$. With $\exp L_{0}^{2}(\mathbb{R}^{2})$ denoting the closure of smooth, compactly supported functions in $\exp L^{2}(\mathbb{R}^{2})$, they showed ([25, Theorem 1.2]) that for all ${\varphi}\in\exp L_{0}^{2}(\mathbb{R}^{2})$, there exists a unique solution $u\in C([0,T],\exp L_{0}^{2}(\mathbb{R}^{2}))$ of (NHE). Continuous dependence on the initial data was not shown so the existence of a local semiflow was not established. In [25, Theorem 1.1] it was shown that for $f$ minorised asymptotically by ${\rm e}^{{\lambda}u^{2}}$ for some small ${\lambda}>0$, there exist nonnegative initial data ${\varphi}\in\exp L^{2}(\mathbb{R}^{2})$ for which there is no nonnegative classical solution of (NHE). Thus $\Phi_{c}(u)\!=\!{\rm e}^{u^{2}}-1$ acted as a critical Young’s function. We consider such cases in Section 7.3 as an application of Theorem C or Corollary C (with $\rho=2$).

Following the template set down in [25], in [37] the authors obtained the analogous results in any dimension $n\geq 1$ for $f$ satisfying

obtaining a unique solution $u\in C([0,T],\exp L_{0}^{p}(\mathbb{R}^{n}))$.

The utilisation of the space $\exp L_{0}^{p}$ in [25, 37] (following its introduction in [55]) is noteworthy, for then one has strong continuity of $S(t)$ in $\exp L_{0}^{p}$; i.e., $S(t){\varphi}\!\to\!{\varphi}$ in $\exp L^{p}$ as $t\to 0$ for all ${\varphi}\in\exp L_{0}^{p}$. This enabled both existence and uniqueness of solutions in the space $C([0,T],\exp L_{0}^{p})$ to be established. By contrast, in several other works (e.g., [23, 38, 55]) solutions achieve the initial data only in a much weaker sense, such as the weak^∗-topology or $\|u(t)-S(t){\varphi}\|_{\exp L^{p}}\to 0$ as $t\to 0$.

More subtly, and of particular relevance here, the proofs in [25, 37] intrinsically made use of the fact that $\Phi(u)\!=\!\exp(u^{p})-1$ is an $N$-function, so that $\exp L_{0}^{p}$ coincided with the Morse-Transue space $M^{\Phi}\!=\!\exp M^{p}$ of $\exp L^{p}$. This is a key observation when seeking to generalise the well-posedness theory to arbitrary Orlicz spaces, since by restricting initial data to $M^{\Phi}$ one obtains both strong continuity of $S(t)$ (via $\exp L_{0}^{p}$) and invariance under $S(t)$ (via $\exp M^{p}$) and thus generate a $C_{0}$-semigroup in $M^{\Phi}=\exp L_{0}^{p}=\exp M^{p}$. This is a key part of Theorem A.

Common to all the aforementioned works in Orlicz spaces are the following two ingredients:

• 1.

  an upper bound is imposed a priori on the growth rate of the nonlinearity $f$;

• 2.

  that upper bound is precisely an exponential function.

These two features have been essential and ubiquitous in the development thus far in the Orlicz space theory for nonlinear heat equations. The reason is that all smoothing estimates for $S(t)$ between Orlicz or Lebesgue spaces relied in an essential way upon some compatibility between the exponential function $\Phi$ (itself induced by the exponential growth bound assumption on $f$ and some educated guesswork) and the homogeneous power law functions $u^{r}$ associated with Lebesgue space norms. This compatibility allows one to estimate the exponential-Orlicz norm of $S(t){\varphi}$ via a convergent series of norms in Lebesgue spaces, each of which can then be estimated by standard smoothing estimates between Lebesgue spaces. A simplistic but illuminating calculation with $\Phi(u)\!=\!\exp(u^{p})-1$ proceeds thus:

from which one deduces (see Definition 2.4 of the Orlicz norm) that $S(t){\colon\!}\exp L^{p}\to\exp L^{p}$ is bounded, with

More delicate variations on this theme lead to finer estimates between different scales of Lebesgue spaces $L^{q}$ and exponential-Orlicz spaces $\exp L^{r}$; see [18, 23, 25, 26, 38, 37] (Laplacian), [5, 9] (fractional Laplacian) or [36] (biharmonic).

We speculate that reliance on this method for calculating smoothing estimates of $S(t)$ is the main reason for limiting investigations to ones in exponential-Orlicz spaces and consequently to nonlinearities dominated by exponential growth. By contrast, in Theorem A and Corollary A the smoothing estimate for the heat semigroup is valid between two arbitrary Orlicz spaces, without requiring any special exponential structure.

For exponential nonlinearities, studies have been concerned mainly with global existence of solutions for small initial data. For Schrödinger equations, the problem with $f(u)\!=\!\pm({\rm e}^{{\lambda}|u|^{2}}-1)$ was considered in [45] in the Sobolev space $H^{n/2}(\mathbb{R}^{n})$. In the context of heat equations with initial data in Orlicz spaces, problem (NHE) seems to have been first studied in [55] where the authors established local existence of weak solutions for small initial data and $f(u)\!=\!\Phi(u)\!=\!{\rm e}^{u^{2}}-1$. Subsequently it was shown in [23] (with $p\!=\!2$) and more recently in [38, 37] in the more general case $p>1$, that global-in-time weak solutions of (NHE) exist for small initial data in $\exp L^{p}$ for $f$ satisfying

provided $m\geq p>1$ and $m\geq 1+2p/n$. We consider such cases in Section 7.3 as an application of Corollary C and Theorem 7.1. There we obtain stronger results due mainly to the flexibility we have in choosing the behaviours of $\Phi$ at zero and infinity independently, rather than being fixed by the single parameter $p$ as in previous studies. Again this advantage is due to the superior smoothing estimate afforded by Theorem A.

Since [25, 37], there appear not to have been any advances on the well-posedness theory of (NHE) in Banach spaces for nonlinearities growing faster than $\exp(u^{p})$. However, some interesting results were obtained in [12] for nonlinearities of unrestricted growth for sets $X$ of initial data rather than initial data in a linear Banach space. As such a solution trajectory does not necessarily correspond to a continuous curve within a space of initial data (i.e., ${\varphi}\in X$ but $u(t)\not\in X$ in general) and does not generate a semiflow. The authors developed a Hopf-Cole transformation method and considered two large classes of nonlinearities, namely those of P-type and those of FP-type. Via an asymptotic quasi-scaling of the underlying equation they established a nonlinear correspondence between solutions of (NHE) and solutions of two canonical forms of (NHE), namely with $f(u)=u^{p}$ and $f(u)={\rm e}^{u}$, corresponding to P-type and FP-type. In particular, they established (among other things) threshold conditions for the existence or nonexistence of positive solutions for nonlinearities of FP-type. Morally, under subsidiary assumptions on $f$ which we do not detail here, if $f$ satisfies (1.4) and $[F({\varphi})]^{-\frac{n}{2}}\in\mathcal{L}^{1}_{\text{uloc}}\left(\mathbb{R}^{n}\right)$, then (NHE) possesses (in some sense) a local positive classical solution. Here $\mathcal{L}^{1}_{\text{uloc}}$ is the closure of the space of bounded uniformly continuous functions in ${L}^{1}_{\text{uloc}}$, with ${L}^{1}_{\text{uloc}}$ being the space of uniformly locally integrable functions on $\mathbb{R}^{n}$. Conversely, for any $r\in(0,n/2)$, there exists positive ${\varphi}$ satisfying $[F({\varphi})]^{-r}\in{L}^{1}_{\text{uloc}}$ for which (NHE) has no positive solution. The work in [12] does not consider questions of uniqueness nor continuous dependence, sign-changing initial data or global solutions. Some of these topics, including global in time existence, were taken up subsequently in [13, 14, 15, 43], again using quasi-scaling techniques. However it does not appear that these methods can be extended to obtain semiflows in linear spaces as here.

We recall some definitions and concepts from the theory of Orlicz spaces, referring the reader to the comprehensive sources [1, 4, 20, 30, 35, 39, 46, 50, 51]. There exist several definitions of Orlicz spaces in the literature, chosen with particular contexts in mind, e.g., functional analysis, partial differential equations or probability theory. It will prove fruitful when deriving our smoothing estimates to choose a weaker notion than those typically adopted in the literature on nonlinear heat equations [9, 38, 37, 55].

It is easy to see that if $\Phi\in\Delta_{2}$ then $\Phi$ is necessarily finite. In the theory of Orlicz spaces, the $\Delta_{2}$-condition is often required only to hold near infinity (usually when the spatial domain has finite measure) while Definition 2.3 is sometimes referred to as a global $\Delta_{2}$-condition or $\Phi$ is called ‘$\Delta$-regular’ (see e.g., [1, Sections 8.6-8.7]).

The most familiar examples of Orlicz spaces are of course Lebesgue spaces $L^{q}$, where $\Phi(x)=x^{q}/q$, $q\in[1,\infty)$. However, note that the Young’s functions permitted by Definition 2.4 include extended functions, i.e., functions which may not be finite-valued. Such functions are not included in the definition of Orlicz space used in [9, 25, 38, 37, 55] for example, but are crucial here in deriving $L^{\infty}$-smoothing estimates for the heat semigroup acting on a general Orlicz spaces. See Example 2.1(a) for the important special case of $L^{\infty}$.

We provide a few illustrative examples for the reader unfamiliar with Orlicz spaces. The dilation in Example 2.1(a) will make a subtle and important appearance in the proof of Corollary A(i).

It is well known (see e.g., [46, Property 1.6] or [50, Section 2.1, Proposition 1(ii)]) that $\Phi^{*}$ is a Young’s function and the following inequality holds:

Recall (see e.g., [6, 56]) that a continuous function $g{\colon\!}(a,\infty)\!\to\!(0,\infty)$ is regularly varying of index $\rho$ if there exists $\rho\in\mathbb{R}$ such that, for all ${\lambda}>0$,

If $\rho=0$ then $g$ is said to be slowly varying. If $\sigma,\sigma_{1},...,\sigma_{j}$ are slowly varying functions then the following are true [6, Proposition 1.3.6, Proposition 1.5.10]:

• (i)

  for any $k\in\mathbb{R}$ and $c>0$, $(\sigma(x))^{k}$ and $(\sigma(cx))$ are slowly varying;

• (ii)

  for any $k>0$, $x^{k}\sigma(x)\to\infty$ and $x^{-k}\sigma(x)\to 0$ as $x\to\infty;$

• (iii)

  the product $\sigma_{1}(x)...\sigma_{j}(x)$ is slowly varying;

• (iv)

  for any $k>1$, ${\int_{a}^{\infty}x^{-k}\sigma(x)\,{\rm d}x<\infty.}$